You don't understand how anaesthesia works, and I think you don't have much personal experience with it?
There's several types.
https://madeforthismoment.asahq.org/anesthesia-101/types-of-anesthesia/
Mainly heavy aka full "put under", then regional/light/monitored, where it's just local and you're "knocked out" if need be.
When theyre doing something like a major internal surgery, (which I've had more than once btw), you're completely put under. No question. You're put actually to sleep before they stern doing anything.
In "smaller" surgeries like this one:

They don't plan on putting you to sleep. They'll make your (in this case) arm go completely numb by injecting lidocaine — much like a dentist. Albeit in this case, it's not your gums they're shooting it to. You know how that's the nastiest part of the dentists visit? Getting local anaesthesia? Well, they don't put it in your gums, they stick a huge fucking syringe in your collarbone (not actually into the bone itself, but the branchial plexus block). Very nasty.
Then they give you fentanyl and whatnot to keep you calm. While they operate. During my previous arm surgery I told them they'd better knock me out. So even with "light" or regional anaesthesia, you get to be knocked out if you need to. And there's constantly an anaesthetist next to you, despite it being only light anaesthesia.
For complete anaesthesia, you're in the bed while they give you propofol and make you count backwards from 10. Then you instantly wake up, having lost x amount of hours. I mean, that's what it feels like. After sleeping your brain/body have a rough idea how long you've been out. With propofol, zero. Nada. Could be 5 minutes could be 5 days. (And Michael Jackson was using that stuff for sleep.) I've been under like that for various reasons.
The scenario you're describing is a complete myth. It relies on there being no anaesthesist there to watch the patient (which there always is with current laws and regulations, pretty much globally I would dare claim) and it relies on the notion of there being some medication that completely paralyses you without stopping your heart and lungs. And that paralysis is somehow completely distinct from pain receptors. You know how paralysed people often explain (well more in the movies but) "I can't feel my legs!", not "I can't move my legs"? I mean, they're not 1-to-1, but after you've been given local anaesthesia for your arm, you don't feel it at all, just like with half your face at the dentist. But you also have no control over it. It's just there.
I compared a lot of things to visiting a dentist, because I assume thats something you have personally experienced?
In the featured picture, my arm was being cut open and operated on while I was awake, but high as balls, talking to the anaesthetist.