There is no need for any Google software. Just install lineageOS, then install f-droid and from f-droid install Yalp Store. From Yalp you can install any app from Google Play, should you want to.
I just looked through the source code real quick, and while I may have missed something it seems to be storing the credentials client side, not server side. Unless you mean on Google's servers, then yeah.
To be fair I don't think it has been available that long in f-droid.
My previous hack was two phones. One clean and one with Google Play. Both with f-droid as f-droid has this wonderful feature of copying one apk to another over WiFi.
Most apps (e.g Signal until today) rely on Google Cloud Messaging (GCM) service for push-notifications. Without GCM, many apps simply crash.