You could have app approvals go through some truly public, government-run agency... except frankly I'd object to that too on free-speech grounds.
Mozilla routinely fails its users and spends a lot of money on downright bad ideas. They are a pinnacle of intransparency and most of their press releases are written in Newspeak, far worse than Google, ironically.
In this very thread we have a discussion about Mozilla putting the boot on its formerly open extension ecosystem.
So it is an odd idea that the manufacturer of the browser (Google, Apple, Microsoft, or Mozilla) should have a say in how much I can trust my content blocker.
The last group of stakeholders I would want involved in decisions about my content blocker is the ad-funded tech industry.
On platforms where sideloading is either impossible (iOS) or usual (Android), adblockers are completely inaffective. The gatekeepers simply don't allow them to exist. This is true even though the maker of iOS ostensibly (!) isn't ad supported.
> I'm also not sure if sideloaded extensions can receive automatic updates
They can't (at least without some external software), which is why the ability to load remote Javascript would be important. For less frequent updates, they could prompt the user.