My Gen1 Fetch TV box went to e-waste recently. It was long EOL and had not worked with Fetch for a long time. I feel your pain!
Tvheadend is great, and you’ll have an easy win paired with one or more USB RTL2832s. I briefly touched on it several weeks ago in this thread. It’s a really easy install, even without Docker. No idea how it would go running on a NAS, but you may find that you need a bit of CPU/GPU grunt to record/timeshift/transcode the MPEG4 TS coming in OTA, especially if you’re decoding multiple streams simultaneously. Some others might have some more hard and fast numbers on the actual requirements, but if you’re recycling existing hardware the best thing is just to try it and see if it works.
I’m not sure what is inside a more modern Fetch TV box, but the Gen 1 certainly wasn’t standard PC hardware (other than the hard drive, which I salvaged for the parts bin), so I doubt that’s a realistic pathway other than building new guts into the case.
It’s been a long time since I’ve run Tvheadend “in production” (so to speak), but if I were approaching it today I’d find something to do the HMI/front end - maybe LibreELEC/Kodi on a Raspberry Pi or equivalent… maybe an app if you’re in the Android TV/Google TV universe (again, Kodi seems like a good fit doing a quick scan of the Google Play store) and something beefy elsewhere to do the OTA reception/recording/transcoding as required (maybe your NAS would be a good fit there?)
I suspect that any sort of smart-ish TV now days will run Kodi (a quick search for webOS + Kodi threw up this article suggesting that it’ll run on webOS devices too) so Kodi frontend directly on the TV + Tvheadend backend on another device might be an easier win than trying to DIY a STB.