In addition to F-Droid on the device, I'd love an F-Droid "package manager client" on a computer, that allows installing apps on an Android device via adb:

    # update repo lists
    fdr update
    
    # upgrade all apps on the connected device
    fdr upgrade
    
    # search packages
    fdr search vlc  # lists org.videolan.vlc
    
    # install on the device
    fdr install org.mozilla.firefox
    
    # download the apk
    fdr download org.mozilla.firefox
    
    # uninstall
    fdr uninstall org.mozilla.firefox 

Of course, all operations would verify the signatures first.