They do, and it's what runs the Intel ME.

Also the Quark X1000 SoCs --- they have an almost unmodified (there are a few Pentium/586-level instructions added) 486 core running at 400MHz. Compare the diagram on page 19 of this:

https://www.intel.com/content/dam/support/us/en/documents/pr...

...with this:

https://en.wikipedia.org/wiki/File:80486DX2_arch.svg

A lot of the text in the PDF above was copy-pasted from the 486 doc too.

Interesting, now I wonder why they don't let ME share a core (or parts of it) with the main CPU. That could impact performance a little, which could be detectable, but I mean, it's a "Management Engine", not a "Spying Engine", right?

They have that idea too in "System Management Mode", which is separate from the Management Engine.

and similarly Atom were mostly pentium on sub 20nm process (with added candies here and there)