If we had to weigh everyone individually, we would never get off the gate.
Sounds like an opportunity to do something useful at the security checkpoint; can't take any longer if there's a scale at the bottom of the pornoscanner, and another in the conveyor to get the carryon items. Just scan the boarding pass and attach the weights to a record in a shared DB that the airline can access. As long as the computer can match a weight to a seat, it should be able to give pretty much realtime balance data.
(If the passenger is only ID'd in that system by either a boarding pass number, or just a simple flight/time/seat string, it shouldn't be a privacy issue, though at worst all you're giving is their weight and that of their carryons. Checked baggage can be weighed at check in.)