> Why is 4:3 such a useful aspect ratio?
I don't know, but I agree with the question's implied premise (4:3's high utility).
It's a good question and I wish I knew the answer to it. I couldn't find any historical reference as to why 4:3 was originally chosen for televisions (the details behind the NTSC format are brilliant, but that's a separate topic).
I suspect it was less because it was "optimal", and more because it was an acceptable compromise between a desirable aspect ratio and technical limitations. Remember, back then, when they were using primitive CRTs, the closer to a perfect circle, the easier it was to manufacture, and most efficient rectangular shape was a square. But humans with their two eyes generally want something wider than it is tall (note movie aspect ratios, which were less constrained by technology). A 4:3 aspect ratio provides something which is close enough to a square to efficiently use the technology of the time, but wide enough to provide a somewhat comfortable shape for viewing.
With non-CRT tech, and modern manufacturing technique, there's a lot more freedom to choose a shape which is good for viewing, so it makes sense there's a lot of experimentation with aspect ratios these days.
Personally I love the "medium-wide" aspect ratios like 16:10 for my main hacking monitor; 4:3 feels constraining. Note that I tend to have multiple windows open (multiple editor windows, an editor and some terminal windows, etc) at the same time, and side-by-side windows are vastly preferable to vertically adjacent windows when the windows are tall (typically true of editor windows). A wide aspect ratio fits this usage pretty well. People whose main mode is the MS-style "one-app-window-always-maximized" may have different preferences.
In the case of the ipad, of course, the main style does seem to be "one app visible", and they strongly want a shape which is viable when used either vertically or horizontally. Given those factors, 4:3 does seem a reasonable choice.