Your post left out PC details but many machines did have a BIOS setting to power on when AC was applied.
Nothing seems wrong here. A new CMOS battery at the shop is under 1 dollar (we buy in bulk) and a few minute job.
So basically, I have connected both my Xbox and my PC to the same monitor and UPS device, but whenever I'm done with my PC and want to get on to my Xbox, my PC will just turn straight back on without me pressing the power button.
Normally what I do is just turn off the UPS device, leave it for about 5/10 minutes and then when I turn it back on, my PC doesn't turn on immediately, but for some reason for the past few days, this has not been working either.
Another thing is, I've recently gotten a message when I boot up my computer saying there's a CMOS Battery Failure, and seeing as the timing of this message matches when I've started having this problem, I was wondering whether this is related.

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