To address the OP's question about why these other aspect ratios not used in the small format line of cameras, all I can say is that they tried that already. There have been several attempts at squeezing more pictures on a roll of 35mm film over the years from your square format (24x24mm) to a slightly compressed frame (somewhere between a 4:3 and a 4:5). In all of these cases, the width of the film remained constant, as they were simply using the 35mm film.
For whatever reasons, those alternative formats just never took off like the ubiquitous 2:3 aspect ratio on almost every consumer and prosumer camera now. It really had little to do with the mirror mechanics or shutter design. That was a solved problem, since all they had to do was make it less wide. In fact, because it wasn't as wide they could probably support faster frame rates for burst mode.
About 60 years ago, they played around with the different aspect ratios. The negatives I inherited from my grandfather proves this as I have everything from 1:1 to 4:2 represented in 127 film (4cm wide). I wish I could point to the place in history where the common 2:3 format became the defacto standard. Now the big companies just don't want to repeat history and risk losing money on something that didn't work in the past.
The shape and proportion of the DSLR sensor format is legacy from 35mm film, which is also 3:2 format. There are medium format digital cameras which use a larger and different proportioned sensors. One example would be Hasselblad H4D-31 with a 4:3 sensor. Another would be Hasselblad 503CWD with a 1:1 sensor. In case you were wondering, the mirrors on the Hassy's are huge, and mirror slap can be a real problem.
Your treasured 4:5 perspective is the realm of large format photographers. There are digital scanner backs for your 4x5 view camera, and these things need a lot of light. It's not quite 4"x5" like the film you would normally use, but it is the same perspective.
On both medium and large format cameras, they use leaf shutters at the lens rather than the curtain shutters near the film you see in almost all (D)SLR cameras. The shear mass of moving a curtain 6cm (medium format) or 11in (really large format) doesn't make sense. The leaf shutters just have to cover the aperture of the lens which is smaller than your 24x36mm curtain in the back of your SLR.
For the folks who like trivia, Kodak's first consumer camera took circular pictures. Getting the pictures developed required sending the whole camera in, and they would develop the used film and replace it with a new roll. The format didn't catch on, as it was very difficult to use artistically. Besides, most paper is rectangular.
One of the advantages of a square format is that you can make it any format you want. You know you will be cropping and will shoot accordingly--but you won't ever have to tilt your camera. Since the square format is currently the realm of medium format shooters, that's a real bonus (those cameras are big).