If you assume you can "sync faster than requests come in to the system", you've necessarily chosen C not-P, i.e. you don't have a distributed system (and in practice you've probably chosen not-A too).
The thing most people pick in practice is CA, single master with hopefully a replicated backup.
That's something lost on a lot of modern engineers, I think. Distributed Systems are something you do because you have to, not because they're the best approach inherently.
The theorem is about being perfectly C, A, and P
If you have more than two hosts, you have P.
If you have delay to sync, you have C.
If you send inaccurate data instead of waiting you have A.
Theres no 'mostly'
so there's a linear relationship between reducing sync time and reducing response time until at some point in reducing sync time you reduce response time below a level you give a shit about...
and you have all 3 of consistency, availability, and partitioning
and then you have beaten the CAP theory