Most major airlines the weight and balance is handled by a dedicated staff and transmitted electronically to the crew.

But even if the crew could do it themselves, they would need to know the final numbers including the distribution of passengers (typically by zone) and the total weights of cargo and bags and where they are located.

In the interest of efficiency, frequently the aircraft is pushed from the gate before either the ramp handlers have inputted the final bag numbers or the gate agent has closed out the flight, sending the final passenger manifest to load planning. This allows the aircraft to be pushed as soon as possible after the doors are closed.

If there is a delay in the process you end up sitting there "waiting for numbers".

If load planning knows that the flight is going to be weight or c.g. critical, i.e. they're trying to get as much cargo on as possible, they will have the aircraft wait at the gate until the final numbers are in. Otherwise, pushing back first and getting underway towards the runway is typically more efficient.