there is no reason why apple or google would need to do this. Rainbow or similar apps just need to change their code in their already deployed wallet software on these platforms to do light client verification.
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.