Afraid it’s not that simple, edge and other chromium browsers can “fake” being Google Chrome as they are based on the same engine. Some websites simply will not load with the gecko engine.
I think it isn't bad to just change user-agent. However, only Mozilla products should be allowed to call themselves Mozilla in user-agent in the first place.
That sounds like a bit of a cop-out, however a Chrome(ium) mode would help me by not needing Chrome as a backup browser (although Edge can do the same thing).
The fact that chromium is open-source means perhaps it could be wrangled into Firefox as well.