On actual server motherboards (as opposed to repurposed home PC's) there is sometimes a special KVM like interface (keyboard/video/mouse, not the VM hypervisor) so you can connect to it with VNC and have the equivalent of local access. This is called IDRAC on Dell servers and other vendors have something similar.
On a home PC, hmm, you might be able to set up some kind of remote power cycle and serial console connection, using a second computer (Raspberry Pi or the like). I'm unfamiliar with Intel AMT that you linked to, but it seems like another idea.
I do remember hearing of a DRAC-like board for PC's but the name of it escapes me right now.
At the end of the day, if you want a long running server, you probably should host it in a data center, maybe with failover and other HA provisions. Home environments are a pain to set up for that. If your computer goes offline and you can't reach it, how do you even know that your home isn't having a power outage? Home ISP's are flaky too, so maybe you want a backup route over mobile data, etc. Yes you can make workarounds for everything but it amounts to turning your home into a crappy low capacity data center.