When I had flat pedals on my main bike (my rigid MTB/hybrid still has them), I wore sneakers or Keen sandals.
A more rigid sole can help with issues like foot pain, so you might try something like mountain bike shoes and just not put cleats on them. The other option might be a pair of something like those Pearl Izumi shoes that are a cross between sneakers and mountain bike shoes. (Look like sneakers, not quite as stiff as a "normal" bike shoe, but still stiffer than sneakers). Or the Keen commuter sandals, but they do run narrow. If you aren't having issues with foot pain/fatigue, then I see no reason to switch from regular shoes. If you want to stick to regular shoes but want something stiffer, my only recommendation is to start flexing soles!