I don't think this is going to help. I have 1847 sent messages. If I would have received 1847 emails, I would have disabled that option a long time ago, or filter them to be deleted.
Better than that, can't we have access to our own IP logs? We get such information from exchanges and even Google so if possible, we can see if any other IP address has been accessing your account.
That wouldn't only work for you, it would also give someone who gains access to your account access to your IP (and approximate location).
I've seen cases indeed where a hacker doesn't change the password so the account doesn't get locked. I'd say that means the account locking and recovery system is starting to pay off!
It's still quite rare to happen, so a solution that can only prevent a small fraction of this rare thing won't really help much.