The amount of repos available to add to F-Droid makes the app selection pretty deep in my experience. Here's a few:

Izzy Droid - https://apt.izzysoft.de/fdroid/repo

Guardian Project - https://guardianproject.info/fdroid/repo

Ironfox - https://fdroid.ironfoxoss.org/fdroid/repo

Futo - https://app.futo.org/fdroid/repo

Fedilab - https://fdroid.fedilab.app/repo

Molly - https://molly.im/fdroid/repo

Briar - https://briarproject.org/fdroid/repo

Anonymous - https://anonymous.gratis/fdroid/repo

There's a bunch more repos you can find online, but I've always enjoyed having these enabled. Additionally, there's the Aurora Store from F-Droid, which allows anonymous access to the Play Store. There's also Obtainium, which will directly access developer's GitHub/GitLab/Codeberg/SourceHut repos as well.

Which apps in particular are you looking for? Aurora Store let's you access the catalog from Google Play, Obtanium can be used to get Apps from Github, developer homepage etc.

APK Pure and APK Mirror do have those apps, but they're not getting installed into my device after downloading. What could be the reason ?

To actually answer your question here:

Are you saying you download the apk file, but it's not installing? So, what exactly did you do after downloading the apk file? I suspect you have to find it among the downloads on your phone and actually open the apk, to initiate the installation.

I hope that helps!

Chances are the apps you're trying to install rely on google shit to work, like their "secure bs" or whatever. If that's the case you'll need to find alternative apps (in the stores mentioned in this thread) if you degoogle your phone.

This might not be your issue, but maybe it is? APK apps need to be installed manually after downloading them.

I use Obtainium as well as F-Droid. Slightly more effort but it pulls from GitHub repos.

Do you have install from unknown sources or the equivalent turned on in your android settings? If so, is your android version too old for the packages you are installing?

I found F-droid annoying after a while, because it frequently lost connection or smth and I got massive toast messages about it covering my screen. I switched to Neo Store, with Obtainium for the couple of apps that are only on Github.

Is your question about app discovery or about installing via apkpure, etc?

izzydroid obtanium komi-store apkmirror aurora-store github

https://f-droid.org/packages/dev.imranr.obtainium.fdroid

And

https://f-droid.org/packages/com.aurora.store for apps from the the Play store when/if you must use them.

I would be very very mindful about downloading APKs from websites.

What app are you looking for?

Do you have an example of an app that's not installing? Which Android OS version are you at?

Try Aurorastore and Zapstore.