Motorola Moto X — Resetting my old phone to give to my son--He has games he doesn't want to lose; I need advice on setting it up.

OK,

Whew, didn't quite no how to word my title! Moving on, here's what I am trying to figure out:

I am giving my old Moto X 1st gen phone to my 10 year old son. He has games such as Minecraft and Monster Legends and he doesn't want to lose his worlds/ progress. So, my question is: I can back up and restore before factory resetting, but every thing is under my Google/Google Play account--and I want to set the phone up under his account and his only, to avoid a confusing device experience. So, if I back up and restore the phone to his account (and not mine)--will I be able to retrieve the app/game data that was there before the reset?

Or, if I back it up to my account, will I have to keep the phone under my account?

I hope that's not confusing. Anyway, I didn't see an option to delete a user under settings on this phone, which would make this simpler.

So, to clarify, all his games are currently installed under my Google/Google Play account. I don't want him to lose his game progress, but I want to set up the phone under his new Google account and I'm worried that he'll lose all his game data.

Phew, I hope that makes sense!

Thanks in advance for any answers.