>>23034490They should just ASK each time.
"Oh hey! We found an egg next to your pals. You've gotta want it, right?"
"<Yes>"
"Ah! You've got a whole team already! Want me to deposit this Egg in your PC for now?"
"<Yes>"
DONE, and possible in a bland, <Y/N> dialogue interface to boot, so I'm sure the crack team at GF could make it work. Eventually.
For exactly one game in Gen VII before backpedaling for no reason.