Do you have an official source? I think the article is wrong.

Author here - I haven't tested using calls with my phone off, my understanding is it should work if you have wifi-calling enabled (from the first footnote here https://support.apple.com/en-us/108300, there's a discussion here where people got it to work https://discussions.apple.com/thread/252472503?sortBy=best). You are correct though that the phone needs to be turned on to send and receive text messages, I'll update the article accordingly.

And I agree with you that AirPods connecting to the wrong device is incredibly frustrating. It happens to me when I have my phone with me when both my phone and watch are unlocked, but it doesn't seem to happen at all when my phone is locked and under my desk. You can also turn off AirPods automatic switching, but not sure how well that works if you want it on between the watch and a laptop but not a phone.

My experience matches the articles calling claims, though I don't intentionally try to go without my phone I just left it at home on a work trip once and it lost charge halfway through.