For the regular OS @fp1_wo_sw_updates already pointed out some of the main considerations.
If you talk about the Open OS, you don’t need to root it again after updates, as it is rooted by default. For the Open OS there was a lengthy discussion about what it should include, where also F-Droid as part of it was discussed. You may want to have a look at the discussion:
Maybe you’ll also like this thread, which is a feature request for the FP (Open) OS (note: this is just a community wish list, nothing official):
https://forum.fairphone.com/t/fp-open-os-feature-requests-vote-for-your-favorites/12303
Anyhow, there you can vote for F-Droid to become a system app in FP Open OS.
Btw: There’s no need for an app to make F-Droid a system app, you only need to copy the apk file into /system/priv-app 