It's been a while since I checked in, but here's an update.
I gave up with EyeTV 4.
I bought an old Mac Pro 3, 1 running Mac OS X El Capitan (which is showing its age with latest standards / browser etc. so I used the dosdude1 hack to upgrade it to High Sierra (the last Mac OS which supports 32-bit apps). I added a cheap WD Green 240GB SSD to boot from prior to installing the hack. It worked fine (if you follow his clear instructions).
Installed EyeTV 3 (courtesy of a backup copy I had on an old drive), ran set up assistant etc and it works effortlessly (as it always did). WOO-HOO!
I was curious about the recordings I'd made with EyeTV 4, so I copied them onto a USB stick and pasted them into EyeTV 3 on my old Mac Pro.
Recordings in EyeTV 4 can be pasted into your default EyeTV 3 folder. Once copied over, start EyeTV 3 and the recordings show up in the 'Recorded' section. They can be edited and exported without issue.
Exports from EyeTV3 once edited and exported, then copied over to my 5K iMac and added to TV App, play absolutely fine. No issues. Image quality from EyeTV 3 HD recordings is really good.
I know this is a less than ideal solution, but it works.
Obviously you don't need a Mac Pro to run EyeTV 3, any Mac limited to High Sierra will do, but the 8- Core Mac Pro I got (specs: 2 x Xeon 3.2GHz Quad Core, 20GB DRAM, with nVidia GTX 680 2GB DDR5 Graphics card), was surprisingly cheap.
There have been an update or two since I last opened EyeTV 4, so I will install them and see what improvements (if any) there are (unless anyone can let me know here if exporting / editing works now?).