Placing a NUC into an eGPU chassis with an MXM card is a novel idea. But, a proper eGPU needs to have it's own power source. If you tied that GPU together with the CPU via a single PSU, well... then... you'd no longer have an eGPU chassis, would you? You'd have a PC. As Aibo pointed out, Mini STX is really the better candidate there.
No, let's not strip the eGPU of its major advantage. One GPU for many machines. Sure, the device you're describing would be incredibly small and efficient, but it's also incredibly niche. I'm concerned a device like that would run the real risk of embracing all of the compromises that NUC and MXM boards represent for the sake of size: High price, low availability, and capped performance (soldered CPUs and GPUs limited to 100W).
TB3 technology is spreading fast. Most of the big players are taking GPUs forward by thinking "Outside the box." Asus, MSI, Gigabyte, PowerColor, Acer, Dell, and Razer are all releasing eGPU products this year. While those products are for full-size cards, all of the most popular cards have a mini variant. Until we have a Mini-STX board with TB3, I think Mini eGPU over TB3 makes a lot of sense for quite a few folks. At least until PCIe 4.0 and OCuLink take hold.
Sorry this response is late. Thanks for your criticism and your views on that idea, I hadn't thought about it from those angles; it was just a neat thought I had.
Also forgive me for not giving more info. I do realize what the purpose of the eGPU is (hadn't thought about using it for multiple machines though, I'll give you that), however at least half of the users of an NUC don't necessarily move them around either (this is just speculation on my part, could be more or less) and most likely won't be moving an eGPU around either, so why not enclose them both in a case and, just as you said, build a single PC with them?
Frankly that would only be for those who couldn't/wouldn't sacrifice a little more space for a proper ITX or STX PC, but the idea is out there for anyone who wants to play with it. Neat idea sure, but not practical for the most of us.