Thanks for the more detailed pictures. Now we can see that the rotary switch has 12 pins soldered to the circuit board. Yes, this certainly is a challenge to desolder on a double sided circuit board without damaging both the switch and the copper traces.
The component ID ( manufacturer and part no. ) is most likely written on the hidden side towards the circuit board, so it has to be desoldered in order to identify it.
Pro repair shops have special desolder stations to do this. Actually I have considered buying one of these to have in my hobby work shop. I repair a lot of electronics, and could really have benefit from this tool. I have to decide, whether it should be a desoldering gun or the more advanced and expensive desoldering station. I'm working on it, and I just found this interesting review:
Anyways, J7373773, the workaround to the issue, suggested by Burner632 in his thread, is a cheap and fully safe way to make the stove work again.
I would recommend the speed control based on a rheostat ( variable power resistor ) for several reasons: It is cheaper, simpler, and more silent. Triac based controls produce some harmonics that can be rather annoying at some settings.
The only disadvantage with the rheostat control is that a small amount of heat is dissipated in it, when turned down.
But remember, the first generations of Whitfield pellet stoves used rheostat controls.
Link to rheostat control, $8.67 on Amazon:
Amazon product ASIN B07W4T5LBH
Link to Varispeed triac based control by Fantech, $25.67 on Amazon:
Amazon product ASIN B000X4PMRY
If you have any questions on the installation, please let us know.
Good luck and take care. Keep us posted on how it works out.