The modern Intel socket CPUs are more like application accelerator cards for this hidden "mother" core. Until recently, you could think of your Intel computer more as an ARC[1] PC with an embedded Intel support slot, than as a pure Intel PC. If you wanted to, you could write a custom BIOS that would boot a Haswell-era motherboard with no CPU inserted, and it would run just fine—because the motherboard already ships with all the CPU it needs to act as a functional computer. (And with the hack on the article, you could run your own software on such a PC, too!)

[1] https://en.m.wikipedia.org/wiki/Synopsys#ARC_International

So no, you can't really remove the ME; it functionally "is" the computer in a lot of ways.

Of course, you could remove its ability to run software other than its kernel—but this doesn't make it "brain dead" or "switch it off"; it just makes it do very slightly less. There's still a lot of (necessary!) stuff going on in there.

Has anyone actually done this? If you have a link to someone who's run a board using only the ME processor I'd be really interested to see it!

I thought the ME was covered in all kind of NDA and proprietary goodness. Booting a board without the main x86 installed would be quite a feat of reverse engineering!