I just used Swift Backup with Shizuku to move an app from my old phone to the new one. It had an apk, but had some other files with it too. I don't know enough about it to know whether the app can be used without those files, or if they're linked to me at all, but the app works on the new phone :)

"Aurora store" is a client for the Google Play servers that lets you download the apk to a directory

Google has been pushing developers toward using app bundles instead of APKs. The actual APK you get is specific to your hardware.

The easiest way to get an APK is to ask the developer. If that's not an option, you can consult a search engine—there are plenty of sites that distribute them (though there's no way of verifying their integrity).

Do you have permission to upload the APK? What app is it?

You should probably check if it's already up on APKmirror. It likely is.

You can use something like this to extract apks https://github.com/AbdurazaaqMohammed/APKExtractor

There are tons of tools which can extract the installed apks from your device. Neobackup and totalcommander for example.