> makes a note of the current time

Makes a note where exactly? In a data store? That means that I'm allowing an untrusted entity to trigger an action that requires me to store and later query my data store. That's a bad idea. The whole reason to have a captcha at all is to stop bots from overloading your system. The data store is a major bottle neck in most systems.

Proof of work is stateless. It's fast to verify. If you sign the challenge input before giving it to the user, you can also statelessly verify it's a legit challenge. No data store needed until after verification is complete.

Edit: also what ahsima said! The point of a captcha is to make it more expensive to use a bot net against you. Timers don't do that.