How does this stove work?
Here's how it works. It's actually simpler than you're making it out.

That dial that goes from 0-5 is a crude temperature controller. It contains a bi-metalic strip which pushes on some electrical contacts. When the temp is low, the strip pushes the contacts together which completes the circuit to the heater. Once the temperature increases, the strip bends away, breaking the contact and turning off the heater. When you turn the dial it moves the contacts slightly, increasing or decreasing the temperature at which this happens.

The light is in parallel with the heater, so you can see whether the element is on or not.

Those 2 wires that go into the surface are the heating element fyi. You're not going to find any active electronics or thermistors anywhere.

Edit: Side note: Make sure that wire that you replaced is rated for high temperatures. It's going to get HOT inside this cavity. You'll notice that all the other wires have high temperature fiber insulation.