This thread has been derailed with low quality answers to the question and off-topic discussions. Part of what makes a forum useful compared to real time chat is the permanence and search friendliness. I've removed the off-topic discussion to make a high quality answer more visible and to avoid needing to delete the thread.
It was fine to mention that Protonmail's app works without Play services but requires it for push notifications, but the thread isn't meant to be a discussion about different email services. Suggesting using the web app instead comes with substantial drawbacks for the already problematic encryption approach and isn't an equivalent alternative.
On GrapheneOS, the Google Play apps are sandboxed apps like any others. Play services has absolutely no access beyond what other apps can access. Every app using Google Play uses the Play SDK and Play libraries so the Google Play code is already part of those apps with or without Play services. Whether or not you have Play services, Protonmail's app includes the Play SDK and the FCM client library. This library doesn't provide a fallback to function without Play services present. Many of the other libraries such as Google's Ads SDK provide fallback code and work without Play services being present.
User installed apps can never access each other's data and they cannot access profile data without permission. They're always sandboxed in the standard app sandbox. Play services and the Play Store are not exceptions from this. Apps within the same profile can communicate with mutual consent on both sides. Google Play being present isn't relevant to whether apps include the Play SDK and Play libraries. Apps can use Google services without Google Play being present. You can see that Google Maps completely works without Play services with only a few missing features, as a nice example of how it's not required. The Google Ads SDK works fine without Play services too. Thinking that not having Play services means apps aren't able to use Google code and services is very wrong. Play services on GrapheneOS is a regular sandboxed app. It has the same app sandbox and restrictions as every other app. If you install an app like Discord using the Play SDK, Google Play already has code execution within the standard app sandbox through Discord. Whether or not you install Play services for Discord to be fully functional is up to you, but if you think you're not giving the Google Play code as much access by not installing it you're wrong and are misunderstanding the whole sandboxed Google Play approach.
Please read https://grapheneos.org/usage#sandboxed-google-play. I think we do a good job explaining it there.