But strictly from an audience perspective, from what I've heard, the live sound suffers with a stage amp.
Often, yes, but not necessarily…
Stage volume takes away some of the FoH engineer’s control since there’s more “noise” competing with the PA. WRT traditional guitar cabs specifically, the main problem is beaming. I can’t count how many times I’ve heard the guitars sound great standing next to the guitarist, but be an absolute ice pick to the ear when I duck my head down to the level of the audience’s heads.
Once that problem is solved, though, for the general
audience (and assuming a good engineer), stage noise is at best a wash — depends on how close the stage mix is to the PA mix.
However, for the portion of the audience who’s right up against especially the center of the stage, they’re not really hearing the PA that well anyway, unless there are “fill” speakers covering just that small area. Good stage sound at an appropriate volume (and there’s a lot
to unpack in that) can mitigate the need for fill speakers to some degree. Enough to maybe not need to incur their added cost and complexity, depending on the situation.