That's the idea.
The reality however is that will never happen unless both Apple and Google are fully signed up to Web3 and incorporate the clients as part of the OS.
And I don't see a single reason why they would do that when it would compromise their billion dollar payment revenue streams.
But this idea that it would be a persistently connected service running in the background on your phone. That's a completely different story.
Is there some benefit to users from OS integration eventually? ie "Pay With Ethereum" in the same way Apple Pay / Google Pay / NFC works, does that benefit from running a light client?
Sure, maybe the incentives aren't there because Apple wants Apple Pay money but whoever does implement it will make their device more useful.
No thank you, though your body and your choice.
Simpler to just change the metamask RPC provider to something I trust or even my home node. Is more difficult to change full-bodied web3 apps which rely on specific data graphs.