I wonder: what is the point of having a manifest, if it is not checked against a head count BEFORE takeoff? It’s been awhile since I worked at the airlines, but I thought that was how it’s done.
As I recall, it is done by about a minute or two before the actual takeoff... There is simply no way to keep the passengers in their seats before the doors are closed, and if they are moving around you can't count them ("herding cats" is the scientific term.)
On the other hand, once the aircraft departs from the gate the crew has solid 5-10 minutes before it gets airborne, and every passenger is secured in their seat. Chances of a mistake are small, and in the worst case the airline risks only one unit of free transportation, a sunk cost anyway since they can't sell this seat after the doors are closed. This is cheaper than taking a couple of minutes on the ground, on every flight. They don't buy airplanes to keep them parked.
That's how I remember it, since I'm not a frequent flier by any means (especially in these days.)