Coming from someone who knows very little about phone :
I have only disabled some built-in Google apps at once. My personal experience probably don't have any relation to you but I know some things that might partially answer the question. (My conclusion is a lot of android games couldn't be used without Google play)
If I don't remember wrong, polytopia is one of the many apps that always tries to connect with Google play every time it's started.
When I first try out an Android phone (Without built-in Google play but with built-in playstore, etc) I was also confused that the game require Google play even though I'm sure such app doesn't do anything to the game. And of course Google play almost doesn't do anything with the game (Literally nothing near essential).
Few weeks ago I tried disabling Playstore (And uninstalled the updates) and the only changes that happen is the apps stopped automatically connect to google play. Later on, I re-enabled the playstore (And can't stop it from always running it self) and the apps started to always tries to connect with Google play even if I re-disabled the Playstore (Including the ones that gets disconnected from G-play and requires another log in to be connected)