I've had no issues with F-Droid. What issues have you had?
I work with free software too so this was a very conscious decision for me.
Run open-source software everywhere else though.
I don't believe this issues you're facing come from android being open, but from google's poor design decisions like granting network access to apps by default.
Apple can do open platforms right if it wanted. It doesn't have to do them poorly like google.
That's the impression Wikipedia seems to give:
https://en.wikipedia.org/wiki/F-Droid
So, for my use case: I want to run some proprietary software, F-Droid will be insufficient.
Happy to be proven wrong.
You can add custom repositories in the fdroid app, as I understand it anyone can host a repo and put whatever they like on it.