Currently, ensuring that an airplane isn't overweight when it takes off relies on the cargo handlers loading the aircraft properly and accurately documenting (on the weight-and-balance sheet) what with how much weight was placed where, and on the pilots properly reviewing the weight-and-balance sheet once the airplane is loaded. Even then, there are uncertainties (it is rarely possible to know exactly how much each passenger weighs, for instance). So why don't airports have weigh stations to weigh the planes as they taxi after being loaded, allowing any flights found to be overweight to be ordered back to the terminal to jettison the excess weight?
Aircraft are heavy.
Really, quite heavy.
An Airbus A340, depending on type, has a maximum takeoff weight (MTOW) of up to 380,000 kg.
A Boeing 747, again depending on type, can have a maximum takeoff weight of a shade under 447,700 kg.
Measuring that kind of weight accurately is certainly not impossible, but it's not exactly easy either.
A bigger issue is that it's not as simple as saying that if the aircraft weighs more than some number of kilograms, then it's overweight, and if it weighs less, then all is fine. The distribution of that weight within the aircraft – precisely that "weight and balance calculation" which you mention – matters, sometimes a great deal. The aircraft could be putting an acceptable weight on the wheels, but if the weight is too far forward or too far aft, may still be outside of its controllability limits once it lifts off the ground.
Simply putting the aircraft on a scale and providing the pilots (somehow) with a weight-on-wheels readout as they taxi will catch the situation where the pilot can't add (as in, mathematically add up the weight components that make up the total aircraft weight), but it won't necessarily catch the situation where the load is improperly balanced.
So you'd be adding another fair bit of complexity (which can fail in various ways) to a currently relatively simple part of aviation (the taxiways or apron), possibly giving the pilots a false sense of security, and not solve the problem. Someone would still need to do a weight and balance calculation, at which point the weight-on-wheels reading can, at best, provide a rough sanity check on the weight part of the weight and balance.
Seems easier to just try to figure out ways to make the weight and balance calculations easier to perform, and make random spot checks on weight and balance calculations done (and schedule followup training if the pilots, or ground crew, or whoever else, need it).
It's theoretically possible, but an airport would have to put in weighing pads for each landing gear, that can accommodate all sizes of airliners, and it would have to be in a specific spot for the purpose, which would require everybody to taxi to the same spot to get weighed causing a huge bottleneck (deicing pans are bad enough).
And the airport authority would have to pay for it, when there is nothing in it for them, and airlines wouldn't pay the weighing fees in the first place. And the current methods work fine.
In essence it's a solution in search of a problem.
There is also economics to consider.
The number of airliner crashes that result from overweight or weight imbalance takeoffs is very low, and almost always confined to marginal operators trying to pack in more paying customers, or not paying attention to weight balance.
Takeoff crashes most often occur when there is too much ice on the wings (Air Florida 90), or the flaps aren't set correctly (Northwest 255), or there is a problem with the landing gear dragging (Lokomotiv Yaroslavl, Arrow Air), none of which is directly related to overweight conditions.
Checking weight on every takeoff would be extremely expensive to implement, would impose additional delays on an industry already wrestling with delays, and wouldn't result in a big increase in safety, because very few airliners crash due to weight problems.
A more efficient course to follow is to continue bearing down on marginal operators, and solve a lot of safety issues, overweight and otherwise, since the bulk of air crashes in general seem to originate with them. More crashes will be averted for less money.
The OP needs to research AF90. The problem was not too much ice on the wings! It was iced over sensors in the engine. The flight crew thought they had selected take-off thrust but instead had selected significantly less, and the plane with whatever ice load it did have couldn’t fly with that little thrust. OP, you need to list crashes caused by overweight aircraft to convince us that there is a problem here... nobody wants to pay for a solution in search of a problem. Convince us, please, that there is a problem here. And thanks in advance for your research on this matter.