Is it safe to use tracking hungry apps with TrackerControl?

I want to migrate my game progress from my previous phone. However, I found out this game (Dominations) has 0/10 privacy according to Exodus. Additionally, I believe I would have to login to Google Play Games temporarily in order to migrate my info.

Let’s say I login to Google Play Games , migrate my info, log out, then continue playing this game along with TrackerControl (or a similar app such as Blokada or NetGuard). Would I be able to keep reasonable privacy under this scenario, or do I just need to abandon this game? Also, how much of a compromise is it to temporarily log in to Google Play Games on an /e/OS device? Once I’m logged back out, Google can’t track me anymore, right? Or does the one-time login have additional consequences I’m not aware of? Thanks for your help!