It’s not necessary to have a control on the heat gun - just use the distance you aim it from and how long you hold it there to change the temperature. Without more detailed information about the values you expect to see you can’t really do much more than prove that it goes up and down as the temperature changes. It should not need much heat to see the change occurring, and you need to be careful about overheating the wire covering.
The values I mentioned in the last paragraph are only to give a guide to sort of values that would give sensible readings from the controller, assuming the particular type of thermistor it expects to see.
I hadn’t considered having the thermistor plugged into the board as you did this test, but the way you have done it has the advantage that you have proven the signal right through to the controller board, so that also eliminates cable and connector.
Rather than trying step 3 as a separate test, you could just repeat that test with the thermistor and hot air gun, watching the multimeter to confirm that the resistance is changing at the controller pins, then start a heat cycle and see if the controller display shows the temperature changing as the heat is applied to the thermistor and the resistance reading on the multimeter changes.
If that doesn’t work then the next step would be to replace the controller. But it is worth trying to update the software before doing that, even though that is a risky process. That’s because if it works then you are OK but if it fails you aren’t any worse off - you still need to replace the controller.