Why don't airports have weigh stations to prevent overweight takeoffs?
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 &ndash; precisely that "weight and balance calculation" which you mention &ndash; 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).