It's a wrapper over Google Play not F-Droid. Much nicer interface, you can filter for things like ads/paid/GSF, you can bypass some regional blocks, you can login with Google accounts or use it anonymously etc.

It lets you install Google Play apps without Google Play itself

Aurora Store is a play store wrapper.

Aurora Droid is an F-droid wrapper.

I use Aurora droid. It is much better than just Fdroid. Search works well, updates are easy, and what is that? App pictures and icons actually load?? 😂 It generally is nicer to use than F-droid and you can add many different repositories to it including custom ones

Takes a few days

Works for me, I downloaded the new version directly from their Gitlab. v4.3.5 worked fine as well.

You might be blocking access to a specific Google Play Store domain or IP. I use Rethink firewall and experienced this when I was blocking a lot of Google's domains. I am using Lineage OS.