The way it might work at an airline is something like this.
The load desk prepares the weight and balance for each departing flight. They actually use a computer program that solves all the math, and it is ready at the final keystroke. They get inputs from several sources.
The flight plan / dispatch release prepared for the flight tells them which aircraft is making the specific flight. Each plane has its own basic operating weight and moment. It also states how much fuel is required for the flight (burnoff to destination + alternate + reserves) and if it is desirable to tanker fuel through the destination or carry "extra" fuel. The maximum allowable
takeoff weight is almost never the maximum certificated
takeoff weight. It is almost always limited by performance or by landing weight at destination on short segments.
When the agent is ready to close the door they confirm the passenger count with the flight attendants, check it with the pilots and check the fuel load aboard with the pilots. After closeout, they input this information into the system and the final count and fuel load gets reported to the load desk.
The bag handlers close the cargo doors and report the bag count and into which pits before they move on to the next airplane. Bag count and distribution gets transmitted to the load desk. (probably straight into the computer program)
The airline's operations specialists at the departure airport (dispatch may be three time zones away from your departure point) confirm the weather and runway in use at departure point. Particularly barometric pressure and temperature must be such that the plane will perform as well as, or better than the pre-flight planning assumptions. If all is okay with winds and weather the weight and balance gets closed out and transmitted to the airplane. Domestically, it usually arrives by ACARS.
The pilots verify the runway in use, the temperature and baro, the passenger count, ACM's (jumpseaters) XFA's (extra flight attendants) live animals in the pit, HAZMAT aboard and maybe a couple of other things. They set up the cockpit for departure according to the data on the W&B message: Flap setting, stab trim, airspeed and engine power bugs. These things become takeoff briefing items.
All of this was done with an FAA-accepted "standardized" weight and balance program that makes a number of assumptions regarding bag and passenger weights by season, and passenger seating preferences. Any time a new gadget is added to the fleet, its weight and location are factored in to the BOW
and moment for each plane in the fleet. It is very complex, but each element of it is pretty simple.
There have been incidents where no one sat in the first six rows or some such thing and the plane was difficult, but not impossible to control until they were re-seated. In other type of (non-airline) operations aicraft have been destroyed through seriously bad loading or by weight shifting during takeoff. (Maybe I'll tell that
war story some day.)
All of this sounds like it would take a long time but it normally happens behind the scenes and in such a way that the W&B comes up during the pushback.
Happiness is not seeing another trite Ste. Maarten photo all week long.