First, please watch this video by Zack Arias:
Of course, bigger sensors built with the same technology and pixel count than smaller sensors yield a better image quality. Of course! It captures more light! But bigger sensors are more expensive and require larger and heavier and (you guessed it) more expensive lenses. Everything in life is a trade-off and so are sensor sizes.
Using comically small 35mm cinema film in compact cameras
The only reason we have a thing we call "full frame" today is because Erst Leitz of Leica fame decided in 1913 to use cheap and widely-available 35mm cinema-film in his first compact camera (the Germans call it "Kleinbildfilm" = "small picture film"). Turned on its side (running left to right instead of running vertically as in movie cameras), cinema film resulted in an image size of 36x24mm,comically small compared to what was used at the time. But why do we still use it, more than 100 years later? Because it was a good trade-off between size and quality, making it possible to build pocketable, small and light camera instead of the 50 pound wooden monsters you had to haul on the back of your truck.
Despite the small film size, images shot with 35mm cameras were blown up to billboard size and have been used for magazine covers. Yes, medium format 120-film about double the size was was quite accessible in the days, but that was an edge-case for high-profile work, like fashion shooting in the studio and slow landscape photography. For every-day picture-taking and for sports and wildlife, 35mm film was the market leader and a lot of pictures have been created in this format, despite being the "small" one. Why? Because the quality was good enough, it was cheap and the benefits far outweighed the disadvantages over large format cameras.
And remember: It was cinema film turned on its side, meaning that motion pictures have been shot on even smaller frames of 24x18mm and are viewed on a massive scale the size of a building. IMAX and other 70mm-techniques being the extreme niche and - even here - not used as widely because working with 70mm IMAX cameras is really complicated and expensive.
Image courtesy of Adakin Productions, via Wikipedia
The quality trade-off of switching from film to digital
When digital cameras came around, they struggled for a long time before professionals were ready to make the switch. Partly because the image quality of the first models was far from what could be achieved with a good 35mm film camera, and partly because professionals who grew up with film were used to the 36x24mm image size, wanted to continue to use their existing lenses and were simply familiar with - say - the field of view of a 50mm lens and the depth of field with a f/2.8 aperture. Much smaller digital sensors just didn't allow the same creative freedom as they were used to.
The availability of a professional quality 35mm, now called "full frame" digital sensor took until 2002 with the Canon 1Ds or actually until 2005 with the advent of the original Canon 5D that wrapped the full frame sensor in a compact camera package. The first Nikon full frame D3 arrived in 2007 and they didn't offer a compact format until the D600 in 2012 (source)
Compared to today's technology, those early full frame sensors do not hold up the slightest. Maximum usable ISO was at something like 1600, just as in the days of film, and dynamic range was more or less non-existent. Still, it was a first step and digital photography has many advantages over film as we all know. Film had better resolution and more exposure latitude, but: In most cases the trade-off of digital was far worth it. Just as 100 years ago when people started using 35mm film instead of 50 pound large format cameras. And just as The Professionals were smiling at people using those tiny little cameras in 1913, The Professionals were also smiling at those starting to use those toy digital gizmos instead of real film cameras in 2005.
Let's move on!
And you know what? Only 10 years later The Professionals in 2016 are smiling again at those who go for smaller mirrorless cameras, because they are not full-frame. And you know what? Time will tell. In a few years, everybody will think back at those huge unwieldy full format cameras and wonder how we even coped with lugging them around. Just as we now look back at 35mm film and just as we now look back at large format field cameras. Time moves on. Technology improves. The "full frame" format is not set in stone, it was just an arbitrary format chosen by the availability of cinema film a hundred years ago.
So if anybody says "I wish the new Olympus E-M1 II was full frame", they are missing the point. This is the next generation. Those are small and fast cameras with tiny and razor sharp lenses, blazingly fast image processors, huge buffers, phase-detect sensors spread over the whole image and modern conveniences like Pro-capture mode (recording images before you press the shutter), eye detection and WiFi transfer. If you want a mirrorless camera, but want to stick to full frame, please go buy a Sony. You will have the same large and expensive lenses as before, just with a different mount with a shorter flange distance, so you have to buy everything again anyway.
Why you should choose a camera system without a full frame legacy
Companies like Canon, Nikon and Sony with a legacy of full frame lenses from the film days usually treat small sensor cameras as second class citizens. If you want to do professional work with professional glass, you have to get their full frame models. Not because you need full frame, but because the ecosystem around APS-C is not suitable for professional work. If you shoot with APS-C cameras like the Sony A6500, the Canon 7D II or the Nikon D500, you have to use full frame lenses which then have weird focal lengths, considering the crop factor. Instead of a nifty fifty you have to get a more expensive 35mm, but there will not be an affordable 35mm equivalent option because that would require a 24mm, but that is a niche lens for astro photography and rare and more expensive. Using a 70-200 on APS-C is too narrow for indoor sports and there is no 50-150. Using a 18-55 normal zoom is often inferior in quality compared to the 24-70 of the same system used for professional full frame cameras. Not to talk about the bulk you are carrying around without any use because you are not using a large portion of the image circle anyways!
On the other side there are companies like Pansonic and Olympus developing the Micro Four Thirds (m43) system and Fuji with the X-trans APS-C size mirrorless cameras. Those systems have been built from the ground up with digital sensors and modern requirements like electronic aperture control and focus by wire in mind. They have been created with a clean slate. Without the legacy of more than 70 years of 35mm full frame equipment. If you want to start an investment in a camera system for the future, my recommendation is to go for m43 or Fuji. Which one to choose is a highly individual decision. There are far more important factors than image quality results from a lab test.
Why mirrorless is still a consumer product, 8 years after their introduction
If professionals are supposed to switch to mirrorless cameras, some basic requirements have to be sorted out first. A professional camera needs to be stable and reliable, provide backup facilities, niche features and must be highly customizable. You just can't lose shots when on an assignment because your camera flaked on you, your memory card corrupted or you have to fiddle with menu settings in the decisive moment. Professional cameras need to be weather sealed, have a rugged feel and an ergonomic grip, they need redundant card slots, good battery life, excellent and customizable manual controls, fast and accurate autofocus and they need to react quickly to user input. Please take note: The only requirement I mentioned that is fundamentally different between traditional DSLRs and mirrorless systems is the autofocus. Everything else is independent of the system and does not have technical hurdles in the way.
So why is there no real professional mirrorless camera on the market? Well, nobody has built one yet. There is no technical hindrance anymore for Canon to leave out the mirror in the next model of the 1D-X. They now have dual-pixel AF and they have all the technology needed to build a professional mirrorless camera. Up until last year, the mirror, relic from 100 years of SLR camera legacy, was still necessary because A) manufacturers can simply reuse their knowhow from the film days and B) on-sensor autofocus was not good enough compared to dedicated autofocus-sensors that have been used for decades and require a mirror for the light to reach them before the exposure. But now we do have professional-grade autofocus on the sensor: The Canon dual-pixel AF in the 80D, 7D II, 5D IV and 1DX II, the Sony A6000-series and finally the Olympus E-M1 II all have autofocus systems that can compete with traditional dedicated autofocus sensors.
This technology is barely three years old and the kinks had to be sorted out, but now we are ready. Ready for the next generation of camera technology to reach the professional field. Highly-capable mirrorless cameras with capabilities impossible to the DSLR-world. But if we take the step to a new universe, why do we need to stick with the full frame moniker? Why do we need to keep the bulk and weight or 100 years of 35mm technology? We don't! With the Olympus E-M1 II and the Fuji X-T2, we now have professional cameras using a smaller sensor, smaller lenses and have all the bells and whistles of the modern age.
Image quality is only one factor
Is the image quality of m43 or Fuji as good as a top-of-the-line full frame camera? No, it is not. But neither was the Canon 5D compared to 35mm film and neither was 35mm film compared to 8-by-10 large format film. Is the dynamic range and exposure latitude as wide as full frame or even 35mm film? No, it is not. Of course you can measure which sensor recovers details in the shadows better if you underexpose by 5 stops and drag the slider up in post. But if you need a high dynamic range, please shoot HDR and if you only can use single exposures for a good reason, you still should not make the need for 5 stops of exposure latitude the decisive factor for your camera buying decision.
With an exposure needing a 5-stop correction, other things in your photography probably need to improve rather than your camera. If that is the most important thing for you: Go for full frame or even medium format. If not, don't try to optimize your every-day photography for rare edge-cases. And if you need to have the ultimate image quality for this one shoot? Go rent a medium format. Go big or go home! You don't drive around in a semi truck because you might need it on one occasion and you don't complain about 4-door sedans that they don't have the loading capacity of a semi-truck. Need a semi-truck on a daily basis? Well then go and buy one and don't complain about the 4-door sedan that is plenty good for most people!
Usability, the menu system, the feel in the hands and the lens selection are more important factors for most people when choosing a camera system. Both m43 and Fuji provide me with a solid lens choice and professional cameras designed around smaller sensors. Where the nifty fifty is a 25mm, where you can get a professional wide-angle that is a 7-14, where you can get a 40-150 f/2.8 zoom for indoor sports that is a 4-times zoom instead of the 3-times of traditional 70-200. Where you can get a 85mm f/1.7 for a third of the price and a sixth of the weight of a full frame 85mm f/1.4. No, it is not the same, you cannot get the same shallow depth of field, but you know how often I used my 85 at f/1.4? Never. I maybe used it at f/2 or f/2.8 in real-world scenarios. On m43, I can use my lenses wide open because they are sharp and the auto-focus is a combination of phase-detect, contrast-detect and eye-detect and will more or less never miss focus, no matter how shallow you shoot.
So if you leave the lab and go into the real world, m43 or Fuji are superior systems designed around small sensors. The image quality might not be the same as today's full frame alternatives, but image quality is not the only factor. Getting the shot and telling a story are far more important. And that one time when you wished you had a better image quality because you need to blow something up to the size of a wall, a lot can be done in post. Compared to full frame from 8 years ago, small sensors from today are far ahead and in a few years we will have put the full frame range of cameras in the same niche as medium format digital cameras. They have their purpose, but they are highly specialized tools. The mass-market, even the professional segment will have moved on. We made transitions like this before, we will make them again, let's do it now! And please stop complaining. It is impossible to buy a bad camera today, no matter which brand, system or sensor size.