By the time it took for the heat to unthaw the vent when the burner started, the CO is spilling into the house and game over. The intake is combined (concentric) with the exhaust (on a lot of them )so it would produce CO from a lack of oxygen.
Everybody should have them. Apparently they should be replaced every 5 yrs as they lose sensitivity. I have seen old ones where the plastic turns yellow and look like they are overheating. I would not trust my life to an old CO detector for the little that they cost. Oil furnaces and wood stoves can produce CO also.