Cannot start or update GmsCore anymore

Hello.
I once installed GmsCore app successfully, started it and made some settings.
Then I realised that my CustomROM does not support signature spoofing yet.
So some time passed by. Now I want to start GmsCore again to make some more settings BUT I cannot find the icon to start it anymore. Just disappeared.
So I tried to make a reinstallation/update but this failed after some seconds of installation try. Error message: "Package is in conflict with an existing package with the same name."
But caused by the missing icon I also cannot uninstall this app before.

Now I ask myself how the start icon of GmsCore could have been disappeared although the app seems still to be installed. E.g. in folder /system/priv-app I can find the folders GMSCore and GmsCoreSetup (which I also cannot delete).

It seems as if I am not able anymore to realise project MicroG on my phone now just caused by a disappeared icon. Strange thing.
Could you explain this phenomena, please?
What can I do to solve it?

Thank you very much!

Start by examining the GMSCore and GmsCoreSetup folders under /system/priv-app, then reproduce the missing launcher icon and the installation conflict described in the issue. Done means determining why the installed app is no longer launchable and documenting a verified way to restore or update it.