A quick follow-up to my post of an hour ago:
Geniatech got back to me very quickly with a link to request a refund.
But meanwhile, I installed EyeTV on my MacBook Pro running El Capitan, connected my hybrid to scan channels, and the program guide updated just fine. So apparently the OS was, in fact, the only issue I was having.
I found a likely-looking file at /Library/Application Support/EyeTV/Shared/EyeTVEPG.db (that's the top-level, not user-level library) and transferred it to my Mac Mini, and my program guide is showing up in EyeTV on the Mini now.
So rather than request the refund, I'm thinking I might just set up an alarm to do a once-weekly update & transfer of that file from my MacBook to the Mini. A bit of a pain, but if it continues to work, I don't think it will be that bad.