In this specific context, where the user reported wrong temperatures, burned things & etc.
PID tuning magic rituals are misleading.
PID has nothing to do with temperature readings. It just controls the heater. That is it.
If temperature is unstable, it may be related to PID params, if temperature is an overshoot/undershoot it is much more probably related. If the fan is cooling down the heater faster than PID reacts, maybe it is worth tuning PID with fans enabled.
There is a topic with “How tos”: Discussion/Proposal: small heater control improvements
There is a suggested solution: RFC: Cooperative Heaters predictive control by nefelim4ag · Pull Request #6837 · Klipper3d/klipper · GitHub
As long, as you see PWM 100% (flat line) (or value equal to max_power) while heating, and see any PWM around the target point, the PID is probably fine.
Your problem is not a PID. As you described, you burn the things several times.
It is either a configuration issue, as pointed out above by @Sineos, or an assembly error.
The real question is: Why does it not trigger the verify heater error?
According to the logs, it seems like, even in this situation thermistor is still heating fast enough, to not trigger the safety shutdown. Looks like default values ensure pretty soft requirements.
Which just does not cover your situation.