I think it would be an improvement to disable (“grey out”) jitsi/integrations/widgets buttons for unprivileged users (per room) …

They are greyed out. If you mean greying out the button:

then that doesn't make sense as you can have personal things like stickerpicker widgets in there too.
What about the jitsi buttons/icons? They don’t make sense if you don’t have permissions to start a call in the room you’re currently in?

It sounds like this is asking for https://github.com/vector-im/riot-web/issues/13118 not the integration manager. The integration manager evidently already does the right thing.