The "thin" in thin client refers to the extremely low footprint. They really are small. And they are a lot easier to set up than a Pi, because they are standard off-the-shelf PC hardware and you can easily add a normal SSD or more RAM. I highly recommend them over a Pi whenever you really want a small PC rather than something embedded with GPIO capabilities. They come with a proper case, various interfaces and a power supply. None of the thinclients I own has a fan or produces any heat I could perceive without using a temperature sensor.
Don't get me wrong – I like the Pi. But only when I want something "embedded" for directly controlling stuff. But there is no benefit when you want a small PC.
For a small machine, you may not even want a graphical user interface / desktop – the command line is plenty for many things. For running Testcontroller, however, I think you need it. Remote access to the machine is trivial: ssh for command line and some flavor of VNC (realvnc / tightvnc / ...) for gui access. It doesn't get much simpler. Browser access to the desktop is quite unusual – don't go that route. Of course, you are right that this will require getting familiar with LINUX to some degree, but that's not rocket science as long as you stay on your local network and don't expose the host to the internet.
So go ahead and stop worrying about the hardware. In terms of OS, any LINUX distribution will do. I like Debian, but Ubuntu, Mint, or pretty much any other will be fine. I recommend picking whatever distro your friends use – that way you get the best local support.