16 batteries at a time, does AA and AAA, automatic shutoff when charged.

The only difference is instead of taking the old batteries to the trash can, you walk them over to the charger. Literally zero extra effort. You state that you're perfectly willing to wait ~2 days for batteries to be delivered, so 4 hours seems like an upgrade. And the charger and price premium of the batteries should pay for themselves pretty quickly.

Bonus: Make it the kids' chore to put dead batteries in the charger and take them out when charged. It's a simple task with a self-fufilling reward/punishment. And the punishment is automatically time-limited and resolved by the kid just doing the chore. Seems like a great way to teach personal responsibility and accountability. Would be totally appropriate for even a ~4 year old (with supervision).

Still not seeing why you think the dash button is a better solution.