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.
No browser as far as I know. I installed the PWA and haven’t touched anything else.
I’m on iOS exclusively with no plans to use a desktop interface