I just ordered EyeTV 4 not knowing that there was an export issue. I used to use EyeTV without issue prior to upgrading to Catalina.
Today after playing around with it some bit, I exported the video in the MPEG-Program-Stream format and then added the .mpg file extension to it. I opened it in Quicktime Player and then saved it as a .MOV. iMovie had no problem recognizing and opening that file.
I'm glad this works but I wish I didn't have to resort to so many steps to be able to work with it in iMovie. I haven't tried importing into iTunes but based on past experience, it should be able to open a .MOV files.
