If that part were broken, then the oven wouldn't turn on at all.
Since it is a simple mechanical temperature switch, it can't be the source of the beep. There's no intelligence in it to signal that it isn't working right, and no way for the oven to test it for proper function.
The switch conducts when cold, and opens when it gets too hot. Since the oven turns on, it is conducting. The only way to see if it will open when it gets too hot is to heat it up. It should not get hot enough to open in normal operation, and you do not want to be playing with that thing when the oven is operating - it switches the main power supply for the oven, and you don't really want to be touching that when the oven is on.
There's something else wrong. The burned wire is a bad sign, but doesn't point to the cause.
This thermo switch is probably good.
The connector might be the problem. It is obviously getting hot. If it has bad contact, that would make it get hot and it would make the oven "think" that the line voltage is too low.
I'd fix the connector, and make sure that there's nothing close to the connector that could have burned it.
You could test it by clipping it to an ohmmeter and heating the front side (with the smooth circular part.)
- When cold, it will conduct.
- If you get it hot enough, it will go "click" and the ohmmeter will read an open circuit.
The trick is getting it hot without destroying it. The barrel of a soldering iron might get hot enough.