If a commercial aircraft's actual weight at takeoff is accurately known (instead of being estimated), can this help with use of thrust, optimal altitude, etc and therefore fuel savings, emissions etc.?
A friend of mine was FO on a regional jet with 50 passengers. During the take-off, the PIC could not rotate the plane at the calculated VR, but only much later. They reported the incident during the flight. The luggage containers were re-weighted at the destination, and it turned out that one of the containers had a grossly wrong weight recorded at the origin.
Weighing mistakes can end in a disaster, especially for smaller jets. Systematic weighing of ready to launch planes would eliminate this risk. I am not aware of an accident caused by this kind of weighing mistake, but surely there have been accidents due to overloading or wrong load distribution.
The technology to build load sensors into the undercarriage is available and it appears that a few planes are equipped with it.
On the other hand side there must always be a risk assessment with a trade-off between costs, likelihood and severity.
[...] measurements need only to be "accurate enough" for the purpose of safety. Being off by small amount would not compromise safety.
For small planes:
[...] small enough aircraft, the actual weight of each passenger (plus luggage) must be used, rather than using a standard weight for estimation.
But, since your question focuses on aircraft performance, we find from the post How reliable is the Airbus FAC calculated gross-weight?
When the A320 is airborne, the [Flight Augmentation Computers] independently determine the aircraft gross weight (GW) by using the aircraft angle of attack sensors. If a significant difference occurs between the FAC and FMS gross weights (delta > 7 tons), the message "CHECK GW" will be displayed on the scratchpad.
That said, when an airliner's weight is in the tens or hundreds of tons, we need to ask how much precision is needed? To the nearest 10 kg? 100 kg?
We see that the Airbus A320 becomes concerned only when the delta exceeds 7,000 kg. That's roughly 10% of its maximum take off weight.
So, the answer is no, current estimations are more than enough for performance [and safety] for the big planes.