I would say Windows Home (as opposed to Pro or Edu) is worse for privacy than whether or not you have Intel ME. If you’re worried about the latter, I heard AMD’s equivalent (PSP) is more benign though I don’t seem to find a clear answer what its network capabilities are.
I believe any Intel processors made after 2008 includes Intel ME or CSME, so I’d assume it does. However, it seems that avoiding ME/CSME (either by disabling/neutering it or by using a super old processor) would do more harm than good.
As for hardware, newer is generally going to be better since they’ll be supported by updates for longer and may include new security features. Old hardware can be especially undesirable if it is no longer receiving security updates.
As for your OS, Windows 10 is nearly about to run out of free security updates. There are ways to continue receiving updates but as far as I can tell, they all involve greater invasions of privacy such as being signed into a Microsoft account. Additionally, I don’t think Windows 10 Home supports BitLocker which is necessary for full-disk encryption.