Even if you're using an end-to-end encrypted RCS messaging app, I have my doubts that they don't have access to your keys anyway.

Depends on. In most cases they won’t see the content of the messages but metadata like when and to whom (IP / Account). I have never registered my device with Google Play Services. Apps like Signal will detect this and will fall-back to their own alternative push notification service which will be established by a background service. Same for SimpleX.

This is why I just switched from frying to find the best privacy respecting FOSS SMS app to just using Googleessages cuz I miss the features that RCS has and nothing is private anyway, except Signal, which I have a really hard time getting people to use