I may be misunderstanding, but which push? The open source project was started in 2014 and was named GrapheneOS sometime in 2019. You may be seeing more about GrapheneOS because of the Motorola partnership, CalyxOS dying/being on hiatus (so many of their users switched to GrapheneOS), and recent news pushing people to use more private OSes.

GrapheneOS won’t be affected. The developer verification thing will be handled by another app and won’t be part of the OS. That app won’t have permission to block app installs or anything like that.

GrapheneOS will be fine without F-Droid.