I have this idea to pick up a used mini-pc such as Lenovo Thinkscentre or similar, to play GOGs on my living room TV. These have lower end specs, but are probably blazing fast for the older games... except maybe for the onboard video graphics processor?

Is anyone doing this? How is it working?

You should check PhilsComputerLab channel on Youtube.

He has done testing on a number of thinclient PCs, including the possibilities of upgrades.

TBH TVs are pretty rubbish for gaming, resolution is not as high as dedicated monitors, and update speed isn’t as fast. Sure for older games it might be ok, until you get RSI and back problems. Especially old games which need mouse and keyboard.
Far better to have a decent gaming setup. Plus maybe a portable device (iPad, laptop, handheld) if you really want to sit somewhere else.

I am and it's great, especially for strategy games and/or controller based arcades. Wireless controllers are a must and a comfortable couch goes without saying.

You should check out Zotac's Mini PC lineup, they offer some great options for this sort of stuff. If you can find a decently specked SH (i5-ish), go for it.

I have a C-series mini-PC and an external GPU linked via Thunderbolt 3, which I hook up whenever I need to play something more demanding. But the iGPU does a good job at powering most of the older titles.

Playing stuff like Disciples 2, Civ 4 or Anno 1404 while chilling out on the couch is IMHO they way these things should really be experienced.

There are some people doing just that, however if you are eventually open to other ideas to be more confused, there is probably better options. From smaller to bigger form factor:

Intel compute stick (or similar 3rd party) is a micro computer that plugs on the HDMI port of your TV. There are some deals going under 50 euros all the time. It's a bit underpowered but has wifi, bluetooth and 10W power consumption. Most versions use the SOC Atom x5-z8350 or the z3735 (wich is much weaker overall).
There is also a development board with the same x5 z8350 unit that goes for under 50 euros, the Atomic Pi. Postage to Europe is more expensive than the board itself but on US and A it should be no problem.

New mini pc using the newer n4100/n4200/n5000 cpu. They are usually barebones and you can add your own ram and storage. Quite more expensive but will emulate games better (if you like it, probably will run Gamecube just fine).
There are some units using laptop or desktop processing units.

If you like DIY stuff, a old laptop will work as well, finding one cheap, like with broken screen may be a challange. This option will have a lot more choice about the performance. For the better looks somethink like a DIY case may be necessary.

Older small form factor pc's are usually much more powerfull than the very small pc's and can take low profile GPU's, opening a lot more options :D
Some will have conventional form factors and power plugs, wich means you can buy a case to fit the living room looks.

Last option, build something new, with a nice CPU like a Ryzen APU.

Keep in mind that the older the hardware, the worst and less efficient video decoders, even if processing or compute power is the same... Some older hardware will have trouble to play youtube at 1080p and very slow browsing the web, let alone 4k videos .

I think a lot depends more on the games than the hardware. Eg, platformers like Limbo & Inside would probably play beautifully, as would many ARPG's and racing games, but I wouldn't want to play games like Banished, Baldur's Gate or Commandos Behind Enemy Lines sitting 8ft away. Ergonomics play a huge role in living room PC gaming for the many older games whose inputs / UI weren't designed around console controllers or 8-10ft view distances. eg, you could buy a wireless keyb & mouse to solve the controller problem, but still not be able to see tiny characters on-screen from 8ft away. If that's a problem, then perhaps one alternative solution is a budget laptop where you can either play up close (on the laptop screen) or plug it into the TV (via HDMI) and use a wireless controller / keyb & mouse for the games that do play well from 8ft away.

4K has been taking over in TV land for quite a while now. It's already getting difficult to find a 1080 model. So no idea in what time capsule you found your information.

I agree on the mouse usage... bad ergonomics on the couch.

Yes, but if I had a grand to spend on a TV with enough resolution and refresh rate to be able to play a game at 6-8 feet away, I would still rather buy my ultra wide 4K gsync monitor running at 144! In fact I would get a second rather than a new telly, most telly in UK is still only just getting to HD so it’s really not worth it.

I use my big flat screen to watch movies and such (bluray, htpc, ...). I can't imagine doing that sitting at a desk.

Telly? Is that the cable thing my parents used to have? ;-P

The newest dell optiplex amazed me the other day. Powerful and complete. Check Phil's Computer Lab on yt for a cheap dedicated gpu for it and you're pretty much done.
If you wanna diy well the Velka 3 case and the dr. zaber ones are a bliss. Silverstone ones (can say personally) are pretty much solid, too. Pc gaming right? Well there's Steam Link for that. Fixed all my problems with it. YES!
Nuff said. PCL is underrated as heck.

Also Linus tech tips + gog themselves showcased a build + some promo games for a discrete pc for the living room.

Blu ray, did that ever take off? My recordings, dvds etc are all ripped to hdd now. Not that I really ever get to watch them. Anyways, stop being ageist, some of us still remember:
https://en.wikipedia.org/wiki/Test_Card_F

Do a search on Amazon (for example).... there's your answer.

I'm not ageist; I hated live TV already when i was a kid. (I gave the "my parents" example because that's the last time I watched that stuff... when I still lived with them) ;-)

True, but then again lowering the resolution to what those older games were designed to be played at in the first place will go a long way. My TV is about 16ft away and I have yet had a problem discerning the UI of any game @ 1024x768. I think it goes without saying that you need wireless *everything* when couch gaming is involved.

While this is not specifically what you are looking for, ETA Prime on youtube has a lot of videos on different hardwares for emulation. Maybe one will suit your needs?

www.youtube.com/channel/UC_0CVCfC_3iuHqmyClu59Uw