I went a step lower than Pentium on my 2 rigs and bought $50 G3930 Celeron processors since I am only GPU mining. They run nvOC quite stably (I just returned from a 5 day vacation and both of my rigs that were running v0016 stayed up the entire time I was gone).
Mine's a Celeron G3920...Skylake vs. Kaby Lake. The motherboard I was using the first few weeks (a Biostar Racing Z170GT7) might not have shipped with a BIOS that supported Kaby Lake CPUs out of the box. That board conked out (was an open-box purchase), so I sent it back and am now running an Asus Prime Z270-AR (only difference between it and the Z270-A referenced in the OP is a lack of DisplayPort and DVI ports, AFAIK).
Granted, I am not using Teamviewer like some folks here. That will consume more system resources. I can do everything I need with SSH and the screen command if I'm at home. I did leave one of my windows workstations online while I was gone so I could teamviewer into that and from there SSH into my rigs if necessary, but luckily I had no need to.
You could configure your router to forward a port other than 22 to port 22 on your mining rig. I haven't bothered with that with mine, though; I can ssh into my FreeNAS media server (or my desktop, if it's booted into Linux...can RDP into it if it's running Windows and set it to reboot into Linux) from outside and then ssh into the mining rig from there. Never used Teamviewer; tried accessing the mining rig with both RDP and VNC, and neither worked. SSH works better for this purpose anyway, once you're familiar with it.