I recall that quite a lot of work was done, once upon a time, investigating the possibility of weighing aircraft on the taxiway with equipment similar - in principle - to that used for vehicle weight checks.
The need for still air was one problem, maybe insuperable.
The idea was to have a check for gross error that could be followed up if necessary. ATC could advise the crew of their indicated weight. It might even be possible to measure the distribution of weight between the MLG and NLG, as another gross check against CoG limitations.
I have always thought it would be a good idea, used for that purpose. After all, it's usually the errors measured in tonnes that kill, not the smaller discrepancies.
EDIT: Out of interest, I just did a search
and found this. It seems I'm well behind the curve, as usual.