I contemplated that for a while a few years back, but decided against it for several reasons:
① I didn’t like an overly-long prompt. On my own machine I already exclude username and hostname, so it’s just `~/Work$ `. (Sometimes PWD gets a bit long, but generally not too much.)
② It was fairly rare that I actually cared about such a timestamp in any way—if I did, it was normally about time taken, and so I could pipe the program through `time` (this was before I discovered `ts` when figuring “someone must have made a program to do this” a couple of weeks ago when preparing to optimise a build process at work), or execute `date; …; date`.
③ Such a timestamp is only written when the prompt is output, rather than when the program is executed (there may be a sane way to have it work otherwise—I didn’t investigate in any way thoroughly—but my initial thoughts suggested any hacky attempt to move the cursor after the fact would interact badly with commands that cause the command line to span multiple lines). This made it useless for well over 90% of the cases when I might have benefited from it (retroactive timing of the duration of a command, rather than at what o’clock I… probably did something).
I return to the desire for a very mildly intelligent terminal that notes when characters were written to screen when not in raw mode, for optional viewing.