This only works if Dave's capacity for responding to email is greater than the amount of email he actually gets. What happens if Dave never wants to spend more than 16 hours per day on email (a reasonable assumption), and his total incoming email volume would require 20 hours per day to get through? Email would queue forever.
For many people, email is an absolute train wreck.
The solution Dave chose is bad because messages get lost
This is the point. Dave would say that his solution is good precisely because messages get lost. Dave is effectively increasing the barrier to receiving an email, and filtering out emails that don't reach that barrier, which is an economically reasonable approach to the problem. (This particular approach may not end up being the right one, but it is worth a try.)