When the simulator loads the AI aircraft, even if an aircraft is due to depart in 1 minute from the time the flight loads, all aircraft need to run through their "Final Call", "Boarding", "Engine Start" etc procedures before beginning their taxi. Therefore this means there will be a delay of 10-15 minutes before you begin to see aircraft depart.
By the time you follow an airliner start -up procedures, as long as there are flights scheduled for that particular time of day, you should be seeing active A.I aircraft.
If you want to speed this process up, A.I. traffic is compatible with accelerated sim time up to 4x. Higher than that will mean the A.I. traffic is disabled and will be reset when going back to normal sim time.
Disabling ground vehicles in the simulator settings will reduce the amount of time an aircraft spends at the gate, leading to faster populating of airports.