Yes, the UPS does power back on when the mains comes back on.
Plug is a WiFi smart plug that I can control from my (another 
) app on my phone. I get notifications when the UPS goes back to AC as well as when HA boots, so if i see UPS is on and HA isnt, I know to go to plan B
And lastly, yes, I could go until the UPS gets to low battery, but historically, if the power is out for 10 minutes its going to be out longer than the 45 minutes my smallish UPS will last, so there’s little point.