What most people miss about CO detectors is that they don't go off until CO levels have been elevated for a while. At low concentrations it can take multiple hours to trigger the alarm; at higher concentrations it can take 30 minutes. Even at dangerous concentrations it can still take 5 minutes to sound the alarm. So you can hold your detector in front of an exhaust pipe for a minute and it won't go off, but stick it inside an upturned pot with a candle inside and it will definitely go off eventually.

The reason for this is to prevent common false alarms like CO emitted due to cooking or smoking.

Depends on the type of unit you had. A "sensor" certainly should have picked up lower levels, but most "alarms" don't go off unless harmful limits are hit - often calibrated at or above 100ppm, depending on the unit specs.