Williams Top-Vent Gravity Wall Furnace won't turn on (Model 2509622A)

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Problem: Burners won't light even though pilot stays lit
Solution: Reset the vent-limit switch (it turns burners off when it gets too hot)

I have a Williams brand wall furnace that was recently installed. Lately it has been not kicking on automatically when the temperature drops. I've scoured the internet and noticed that other people have had the same issue. The pilot is lit, the thermostat is set properly, but it still does not kick on. At first I thought this was due to a loose connection but ultimately what I noticed is that at the top of the unit behind the side vents, there is a "vent-limit switch". This is essentially a switch that turns the burners off if it overheats. The pilot remains lit when this happens. There is a tiny red button that you need to push to reset this switch each time this happens. Luckily you can fit a letter opener or something thin between the side vent slats to reset this button if this happens. The odd thing is that this seems to happen when we lower the temperature so the furnace doesn't even kick on. Hope this helps someone.

Some additional notes...

If pilot will stay lit properly but burner doesn't turn on and it's not the vent limit switch, it is likely just a short in the wiring to the thermostat. You can check this by turning to the lowest temp on the thermostat then by shorting the top and bottom screws on the gas valve. A paperclip should work fine to check this.

Another thing to check is the voltage between those is at least 300-400 millivolts with ethe pilot on