I understand that we use monitors as multi-purpose devices but just because a Swiss army knife has a corkscrew doesn't mean that it can do a better job than a wine bottle opener designed specifically for that purpose. I have no problem with current monitors, I have been a digital photo restoration artist for 18 years and spend a lot of time in Photoshop. I appreciate the extra working space that current displays offer for tools, menus etc. but when I view a slideshow of my finished work, a lot of which is on portrait oriented images, it is a little disappointing to see scaled down versions with huge swatches of wasted space on my screen. Same goes for my own personal photographs. It does not matter if the final output is a print that can be physically rotated for proper viewing but, prints are no longer the primary viewing method anymore.
Since I have to backup all of my work anyway, whether business or personal, I would much rather have the option to view all of my images in the same size even if it takes another device to do it. Not all readers feel compelled to purchase a Kindle even though it offers a smaller, glare free, longer battery life etc. display than an iPad but serious readers seem to prefer them. It does not seem unreasonable to think that a photographer would consider a separate device that provides more consistent image treatment than current devices. I would not watch movies or edit photos on a square screened device but I certainly would rather view my photos on one.
Note that if you purchase a 4:3 digital photo frame you will still have a size discrepancy between differently oriented images, just not as drastic as on a 16:9 display. The same is true when viewing multiple images using the full screen of a rectangular display. Any image that does not match the screen orientation must be smaller than images that do match. Do not confuse the layout of multiple photos on an image editing program which are arranged on a screen within the screen and can be scrolled through with no regard for wasted space because the images do not encompass the entire screen.
Asymmetrical images cannot be rotated on an asymmetrical screen without reducing the image size of one orientation and asymmetrical screens cannot be divided into symmetrical segments. Only a symmetrical square screen can allow an asymmetrical image to rotate freely without resizing the image Only a symmetrical square screen can be divided into symmetrical grids to allow multiple images to rotate freely without resizing the images.
Square screens may not be your cup of tea but, they offer the only alternative if the goal is to achieve consistent image sizing and placement of photographic images.