It seems to me that takeoff weight is an important piece of data, and if one had weighing scales between the wheels and the aircraft body, one could precisely determine it. However, searching the internet, there seem to be no scales and cargo/fuel/passenger weight is estimated by adding up what is brought on board. Why aren't scales common? And are there any commercial airplanes that have them?
Yes, such system exists, for example the Honeywell WBS.* It was developed during the sixties and installed as an option on the Boeing 707-300 freight aircraft. Other aircraft that had a similar (optional) system include the L1011 and Boeing 747.
It is mostly found on freight aircraft because they have a less predictable load distribution (centre of gravity) than passenger aircraft.
The reason they are not common is that they add cost in the form of installation, weight and maintenance. While quite accurate when in good condition, the reliability of the weighing systems used to be poor, which probably contributed to the fact they never became wide spread.
Because crew are commonly operating on several aircraft types these days that have take-off weights of more than 100 metric tonne difference (e.g. A330 and A340, B747-400 and B747-8) there is an increase in take-off weight confusion related incidents. This, combined with the fact that systems are becoming more reliable, makes it likely that these weighing systems will be used more in the future.
* An advert for it in a 1987 Flight issue.
For one thing it will not work because of wind: Once you put wings on a fuselage, you will need to bring that airplane into an enclosed hangar for the scales be accurate.
Mind you, this is exactly what being is done whenever an airplane is being weighted for mass & balance purposes, about every three years: enclosed hangar and massive scales under the gears. Doing it on a daily basis in an already crowded airport is not an option.
What you really need to know for your a/c is mass, not weight. Also, you'd want to know the mass distribution inside the a/c, not just to calculate the center of gravity (which a scale can also do), but also to make sure that load limits are respected in each section (which a scale will not do). Adding and removing numbers on a load sheet works pretty good for this purpose, as does estimating passenger weights and luggage as an average, although there are talks that those averages need to be increased lately.
Even if you eventually found a way to scale each airplane before takeoff, in a controlled environment, the extra costs will never be justified since you basically improved on something that works.
On some newer B747, there is a weight indication near the maindeck door. I seem to remember Atlas Air aircraft having this. This would probably have its sensors somewhere in the landing gear.
It was a big LED thing and showed you the current weight of the aircraft, e.g. 340.5 (thousands kgs). When a pallet came in it would show a corresponding increase in weight, so if a pallet weighing 3000kgs came in the weight would now show 343.5.
It was never used for calculating the weights of the pallets in the airports I worked from but I understand that Atlas does a lot of operations for the military in some dodgy areas and at these airports this device helps verify that the weights are not too far off.