I'm not so sure. I think the privacy-functionality trade-off is understood and expected by the users, it woul be used by very few people if it were extremely spartan.

(But this instance, the favicon service, is not a good privacy-functionality trade-off)

What would a browser even need a backend for? The only valid use that I can think of is Google's Safe Browsing list, but if ad blocking can be implemented totally on-device, surely that can, too?

The use case I was going to suggest was a safe browsing list. Possibly also something like Have I been Pwned. So, yes, there are valid reasons to have a backend but they are very narrow and privacy is key when building them.