Since we are all here in an astronomy forum and, therefore, are probably more well-versed in all things related to telescopes and astronomy, here is another way to look at using a full frame camera sensor on a microscope.
When doing astrophotography, even some 2" focusers have difficulty accommodating the larger full frame sensor. The diagonal of the full frame sensor is at around 43.2mm. A 2" focuser is at around 50mm but, because of the length of the focuser tube running into the wider part of the light cone (forward of the focuser), you end up with a bit of vignetting at times. A 2.5" or 3" focuser gives a bit of extra room for full frame and medium format sensors. The light path is actually a cone so it is easy for that 2" focuser tube to infringe upon the cone of the light path.
Now, back to microscopes, the optical tubes in microscopes are 30mm or even 23mm. Trying to image using a full frame sensor that is 43.2mm across will result in significant vignetting. You will actually see almost the entire optical tube within the frame. 30mm is only around 1 3/16" so that is comparable to a 1 1/4" focuser tube and we know from astronomy that 1 1/4" focusers are not used for full frame sensors. What compounds this vignetting problem, however, is that this 30mm optical tube is fairly long on a microscope so that adds to the vignetting issue. A 23mm optical tube (like I have in my microscope) is around 0.9" so smaller sensors are needed to eliminate significant vignetting.
One way around all this vignetting is cropping but then that is the same as just using a smaller sensor. If you have the number of pixels of a recent full frame sensor (in the 24mp to 42mp+ range), you probably have the ability to crop effectively making the sensor an APS-C sized sensor or a 4/3 sized sensor with no perceivable loss in resolution.
Another way around all the vignetting is using a 2x lens in your adapter for mounting to the microscope. Unfortunately, then you often end up with blurry edges due to the 2x lens being not-so-great. Another option is eyepiece projection. I haven't tried that yet but common sense is telling me a full frame sensor will still vignette significantly with eyepiece projection. Playing with spacing between the eyepiece and the camera might help though.
Video is a little tricker. If you can shoot in 4K, it may sound like a lot but you are starting at only 8mp. Cropping that 4K video to Full HD drops it to only around 2mp. My computers are all displaying at 4K and the same holds true for my televisions. HD video or 2mp images are what I refer to as thumbnails on my screen. They are actually difficult to view. If you stretch them to fit the screen, you only end up with hollow magnification like trying to use too much magnification in your telescope. So, for video, a smaller sensor is preferred rather than trying to crop.
Newer cameras have more options and larger sensors so they are more easily adaptable to circumstances like this. Older cameras are not so easily adaptable. The same really holds true for newer microscope cameras but it gets trickier figuring out which of the microscope cameras have newer sensors unless you follow the sensor business.