Some aircraft use weight sensors. Most do not.
Determining the weight of the aircraft is the function of the pilot in command, or those designated to do it for the pilot in command. The PIC will be the person to finally approve the completed weight and balance calculations.
While various sensors describing the weight and balance of the aircraft have been used, some have been deactivated because of maintenance and problems they cause. The empty weight of the aircraft is determined on a regular basis. Whatever is put on the aircraft must be calculated into a final weight and balance document; the aircraft must not only be within weight limits, but a fairly narrow range of balance, to ensure control on takeoff and in flight.
Aircraft, even large airplanes, are not comparable to heavy mining equipment. Aircraft are relatively fragile, and a mistake made on weight or balance can have disasterous consequences. Overload a dump truck and it operates a little slower. Overload an aircraft and it may never get off the ground, instead resulting in a fireball off the end of the runway. Put a little more load aft in the back of the dump truck, it may ride a little nose high. Put the center of gravity slightly aft and one may not have adequate control authority to fly. The implications of even minor improper loading on an aircraft are far more significant than large scale errors on heavy equipment.
When we calculate our weight and balance, we calculate not only for cargo, people, and other load items, but we calculate for the weight of the fuel, and the shift that will occur in flight as fuel is burned. We have intermediate weights we must attend, to include a zero fuel weight; the weight of the airplane before fuel is added. We have balance issues between fuel tanks, and between cargo compartments or sections of the airplane.
The manner in which the airplane is loaded, not just simply it's total weight, affects the way the wings bend in flight. The strength of the wings and their ability to withstand inflight loads such as gusts, is derived in large part from the balance applied to them and the wing bending moment. These are critical issues, and not ones that are generally left to a meter or sensor on the landing gear.
Calculations are done with all individual factors combined and talied, and carefully placed throughout the aircraft to achieve both a weight that is appropriate to the operating conditions and limitations, and a balance that is safe for operation. We can not simply load the airplane and see what it ends up to be. While sensors on the gear might tell us if the airplane agrees with the final calculations, the loading must be determined at the outset. We don't simply herd everyone on board, pack on as much cargo as we can, and then say "Ah, we need to take a little off."
There may be occasions when a calculation finds an error after loading is complete and adjustments must be made, and there are times when ambient conditions such as the temperature changes between the time calcualtions were made and departure time. Recalculations may find that given an increase in temperature, for example, weight must be reduced for a safe takeoff and climb performance.
Weight must be tailored not only to the structural limitations of the airplane,but also to the field conditions at the departure, destination,and alternate. We may be limited by how much cargo, personnel, or fuel we can carry due to temperatures, runway length, or even enroute conditions that might limit our cruising altitude, cause more fuel burn, or less fuel burn (such as strong tailwinds). These affect the loading, and must also be taken into account. These calculations cannot simply be made after everyone is aboard and the door closed; it's too late to throw everyone on, see if onboard sensors give us an acceptable weight,and go. Generally operations are a little more advanced and technical than that.
Standard weights are used by some operators, and actual weights by others. Every few years the standard weights are revised. The general populace tends to gain weight as a whole, and the standard weights tend to get revised up a bit each time. In unusual cases, each operator always reserves the right to weigh individual persons, if necessary. This is generally done more frequently in small aircraft than large aircraft, as larger airplanes tend to be a little more forgiving, with wider envelopes that can handle changes in center of gravity. Individuals make up smaller percentages of the total payload on large airplanes too; individual excursions from the standard model are more easily tolerated.
Baggage is generally weighed, although standard weights may apply up until a given limit, when a higher standard weight (heavy bag) applies.
The location of personnel and baggage makes a big difference in how the flight may be impacted. On the 747, for example I've seen the movement of a single person from the cockpit to the back of the upper deck make a palpable trim change, and I've seen balance calculations changed under certain loading conditions by moving a single person. Conversely, I've flown some airplanes in which I drop more than half the loaded weight of the aircraft in flight; cargo, jumpers, chemicals, etc, without an appreciable threat to the control of the airplane.
It's far too simplistic to simply think about throwing an electronic scale or meter on the gear and calling it good. Every detail of the load must be accounted, whether it be a passenger, or a case of oil in a lower cargo area.