It's called a sail switch and has multiple functions in relation to the heater's operation. You are correct in that it monitors air flow. The fan should run a little while before it lights to clear any leftover gas and a little while after the flame shuts off to do the same. The sail switch tells the control board that air flow is present. The process starts with a 'call for heat' and ends when the thermostat 'thinks' the desired temperature is reached. What happens in between is an article in itself.
The sail switch could be be bad or it could just be corroded connections. Check the 12v supply going to the control board and clean them then test. The control board can be bad and this is a common problem; particularly where moisture may be a problem. It's a very expensive thing to replace so save it as a last resort.
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