It is not a converter - it is a miniaturized T12 controller integrated into the handle. The addition of the diodes prevents reverse voltage damages.
Anybody tried this analog controllers for T12?
It is a proven design and many copies have been built. So yes, it will work.
it's an oscillator whose duty cycle is controlled by the thermocouple voltage
Depending on your definition of "oscillator".
If the thermalcoupler's output is too low, U1A's output goes low -> U1B's output goes high, turning on the heater -> for a period controlled by R4/C2. After that, the heater is turned off, the temperature detection phase begins again.
The diode on U1A does all the trick.
If the