F-Droid is an installable catalogue of FOSS (Free and Open Source Software) applications for the Android platform. The client makes it easy to browse, install, and keep track of updates on your device.
Something worth noting is that F-Droid is both an app to download other apps but they also maintains a repository of apps. You can use alternative store apps (like Droid-ify) with the F-Droid repository OR you could use the F-Droid app with a different repository (like IzzyOnDroid). You can mix and match to meet your needs.
I use the Droid-ify app with the F-Droid, IzzyOnDroid, microG, NewPipe, and Collabora repositories.
Once you start down this rabbit hole, give Obtanium a look.
On Android, we're used the "Play Store" being both the app that facilitates downloads as well as the collection of apps available. With F-Droid, you can add additional collections of apps to make available for download.
You might add an additional repository to gain access to apps not in the main F-Droid repository. You might add a developer's repository to gain access to updates to their apps before those updates hit the main F-Droid repository.
Divest is the developer repository for app maintained by Divest OS, a fork of Lineage OS.
Some software developers prefer to host their own repos and have more control over the release process and/or don't want to fill all the criteria for being included on f-droid, so they create their own repos. Some of these apps can still be found on vanilla fdroid, but often aren't updated so frequently.
Izzyondroid, on the other hand, is a different project, aimed at hosting different apps that are usually from smaller devs and can't be included on fdroid yet, for different reasons.
The greatest thing about fdroid is that it allows anyone to create their own repos and you aren't forced to depend on anyone.
There is safety there, but you're just as safe using the the developer's own repository for their apps, like NewPipe, Collabora, or the Guardian Project.
Many years ago I tried to go completely de-googled, and that involved using only F-droid. One of the many problems I faced was the tedious update process. I needed to tap each and every app individually every time there were updates. I wonder if droid-ify could have fixed that. Unfortunately I didn’t come across that app at the time, so I didn’t try it out.
Oh for sure! Droid-ify offers a few different installation methods. The Legacy and Session install options are what you are used to. With those methods, you are prompted to download and install with each update.
With the Root install method, updates can be downloaded and installed in the background using root privileges. Lastly, and I think most intriguing, is using Shizuku. Shizuku is a utility that will give you close to root access using ADB. See link for details. So, with the Shizuku install methods, Droid-ify can keep all your F-Droid apps up to date with little intervention from the user.
Footnote: Because Shizuku leverages ADB, it needs to be started manually after each reboot.