It would be possible to design a system that computes an airplane's weight and C of G from the landing gear load in real time, either by incorporating load cells into the gear legs, or measuring nitrogen pressure in each oleo (accuracy might be a problem there), and processing the data with an on-board computer. Or you could just include weight sensor pads on the tarmac somewhere, and weigh and compute the loaded airplanes as they taxi over them like trucks at a weigh station.
Problem is, you need to be able to forecast the all up weight and C of G beforehand anyway, to plan the load amounts and load distribution, fuel load etc. An on-board load sensing system only helps you after everything is complete.
Since you know the weight on the gears of the empty airplane before you started, it's just a simple math calculation to work out the numbers, and being able to do it beforehand allows you to adjust and tune your loading. It has a long history of being an accurate and reliable methodology, and incidents are too infrequent to justify major reform of it.
At the end, because a system to measure and compute the load on the gears in real time is kind of redundant, nobody is going to want to pay for such a system with the cost of certifying it.
So basically, there is no market or regulator demand for such a system. If a rash of incidents occurred that made it clear that the existing methods were unsafe, then you would probably see a move in that direction and start to see development of that sort of system.