I'm thinking about doing a similar setup and I like the smaller form factor because I can attach it to the back of a monitor and not even have to look at it.
Take a look at the Antec ISK-110. Not as small as the NUCs, but lets you build your own machine on the cheap, and still designed to be attached to the back of a monitor.
(Won't fit a standalone GPU, but I don't think anything that can be attached to a monitor can)
(Won't fit a standalone GPU, but I don't think anything that can be attached to a monitor can)