Certainly in all HOUSES I am familiar with, there is a visible
tank (and usually, but not always, the stage). I'm not sure
what the original motivation was. Maybe it cuts down on water
usage. It makes backsplash less of a problem as well.

Thinking about toilets in public buildings etc, yes, now that I
think of it (this has previously not been on my list of things
to notice), visible tanks seem to be the exception. Maybe there
is some system with high-pressure water in pipes, valves etc which
doesn't need a tank? After all, the idea of the tank is to provide
pressurized water for the flush via gravity.

Let's start a thread on bidets! :-)