The darndest thing...
I uninstalled EyeTV with EyeTV Reporter, then reinstalled it again. Nothing happened.
Then I installed EyeTV 3 in a virtual machine (Mojave), so I could at least watch my favourite shows. This worked, but wasn't very stable, so I returned to EyeTV 4 (on Catalina) and now it worked.
I'm guessing, the switch in USB ownership from Catalina to Mojave and back did the trick. (I'm using the TV2 Hybrid USB stick). The USB interface was a bit messed up after the upgrade, dismounting disks again and again.
