TinyMDM (and some others) have quite slick app management including integrating with Google Play Store for both private and public apps where admins can search and browse apps to add to the approved apps list from within the mdm portal.
There is a this 5 year question about how to install from the app store:
https://qa.h-mdm.com/1030/how-to-add-an-application-from-the-google-play-store?show=1030#q1030
Adding apps by play store or fdroid url would work quite well too.