Just invent a chicken pokemon that sits on eggs. Every daycare has them. You put your eggs under it, and it dramatically increases the hatch speed. Then streamline the process so that the pokemon keep laying them and the daycare keeps putting them under the hen without having to babysit each step of the process as the player. You get phone calls after the first egg and then after each set number of eggs going forward to remind you. Then you get your in world flavor, and make the process less annoying and tedious
*ring ring*
"Your pokemon had some eggs! Would you like us to put them under Motherclucker until they're ready to hatch?"
"Sounds good. We'll just keep putting any more under there too. They should be ready to hatch in no time!"
a reduced number of steps later
*ring ring*
"Your pokemon eggs have hatched! Please take whichever ones you want to raise, and we will take care of, find homes for, or release the rest for you. Seriously, don't think too hard about this or make it weird, just abandon whatever you don't want. It's fine. We aren't going to cook and eat the leftovers or something. That's just a nasty rumor being spread around town."