When that person goes to the cold storage wallet, the only trustworthy information they have (assuming compomise of the online system) is how much money they are transferring. This means that this arrangement succeeds at making a trustworthy rate limit and nothing more.

If I wanted to make a trustworthy rate limit, I'd have the machine from (1) not airgapped, but connected by a serial (unidirectional) line to the machine from (2) along which it would periodically send signed transactions sending some of the money from "lukewarm" storage to hot storage, and sending change back to the very same "lukewarm" storage address. The machine from (1) can then implement a rate limit.