Ok, for anyone having similar issues, do this:
#1. Remove Lutris flatpak
flatpak uninstall -y net.lutris.Lutris
#2. Remove Lutris flatpak .desktop file if it was not removed. It interferes with proper RPM one somehow, GNOME magic
rm ~/.local/share/applications/net.lutris.Lutris.desktop
#3. Log out & log in back, now you should have your proper Lutris launcher