I'm guessing it's not that simple or that's how it would be done, but if my seat can weight how much I am so it knows how hard to blow the airbag up in my face, then surely it can scale up to weighing how much load is on the suspension of the plane?
Some can, but it's not accurate enough for reliable use. Sloping aprons, uneven tarmac, and wind will all throw off the calculation. The aircraft can calculate its own weight pretty accurately once airborne using aerodynamic data.