I seem to recall a suggestion from years back similar to kicking eggs to the AP. But instead of putting them in the AP, sending them to the back of the queue so whatever's behind it can be seen.
Still, I do remember the mad dash for the hourly drops when there was only one cave. I also remember cave blockers and the cave stagnating when nobody wanted to lock themselves with the purples (or frills). So the problem persists regardless. I think unlimited egg slots could work if you limited the amount a person could pick up at any one time, which seems to be people's main complaint. For example say you snap up 10 eggs in the hourly drop. Fine. But then your scroll's in a cool down period. Five hours later, you can hunt again and keep your previously caught eggs. It at least hinders people from taking too many eggs at once and increases the chance for you to see something you really want.
I like a lot of the compromises here too. So whatever would work best. Honestly, I'd slap my mother for a viable (read: permanent) solution to the cave moving at a snail's pace.