Presumably in that case they could avoid paying you by canceling the non-compete. They could decide whether the information you learned during your one month is worth paying you.

That does sound nice. Ideally though the company wouldn't chose to enforce the non compete because the new employee hadn't picked up any secrets worth sharing in a month.

Actually, this might incentivise a new employee to hunt for secrets so they could quit as soon as possible and enjoy their serverance. That might not be ideal.

Even in the absence of non-compete clauses, you're still obliged to keep your company's trade secrets when you leave, or they can sue you. California companies sue each other all the time over things like this.