A case that doubles as the heatsink ... I like the idea. Reminds me of Confederate Motorcycles - they like to make the frames of their bikes more functional - the gas tank and oil tank are a part of the frame (with even cool side windows to see the oil and gas sloshing around inside the frame)
Anyway ... back to the tech world, I love the concept of making everything multi-functional as possible.
I wonder though, how warm the case might get after some heavy load - hopefully cool enough you can still comfortable plug and unplug peripherals? I could see it potentially getting a bit toasty.
That cheapish base model could be neat as a silent, more flexible option to drive a TV (vs. the usual sticks and set-top boxes).
Re: quiet AMD STX machines, I'm writing this on an STX DeskMini A300 with a 3200G's stock cooler (with the shroud taken off so it can be crammed in there) and the "Quiet" BIOS setting. I can hear it if the room's otherwise silent. But it's quiet and consistently so! The NUCs I used before revved the fans with minimal provocation (animated GIFs in work Slack ). Although this chip has a 65W TDP, in practice it seems like it takes heavy multithreaded/GPU load to stress the cooling. I'm not gaming; that would probably push it harder.
The less sketchy way to do something like this is probably to get a low-profile Noctua cooler or such; I used the stock one because I had it and was cheap and impatient and it (barely) fit.