It is a limitation of a PWA. A native app (wrapper) will likely be coming though, and a custom Capacitor plugin could theoretically interface with the native API to switch out icons. Theoretically. No promises though :)
Quoting this comment by the dev
Problematic
I get lots of random freezes, actions not registering, etc
Doesn’t make it unuseable, but it’s not a great experience
With all browsers?, here runs fine with chromium.
The main issue is Safari, bugs in webkit bog everything down. And iOS is stuck with it.