Well, not quite thousands of variables. More like about a dozen.
Before microcomputers came along, we used slide rules, like this load adjuster that was custom built for each cargo aircraft model:
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You'd add up the weight in each compartment or at each station, and then it graphically showed if a load would be out of balance for different parameters (such as remaining fuel).
The other part of a loadmaster's job doesn't require calculations, but knowledge about how to tie down cargo and prevent problems with hazardous materials.
Cool antique!
It's not just location of loads in the C-17, it's also abilities like air drops.
The floor has 12 to 20 remote controlled release latches for automatic release in air drops. It has all the equipment on board to convert into various mission profiles. Much more to take care of then a C-47.