Actually, the trim sheet is rarely "correct" as it is entirely routine for the designer to build in "design errors" to account for "usage errors" .. with an expectation that the two cancel out on most occasions so that, regardless of what the sheet might say, the aircraft loading has a high probability of being within the TC limits.
There is a basic presumption in the use of typical manual loading systems that there are checks and balances to guard against gross finger trouble errors .. certainly doesn't guarantee the absence of error but, if sensible and used, the overall quality of the manual system ought to be reasonably good.
Some examples of checks and balances are ...
(a) having a crew member check the can weights and order of loading AFTER the load is on board (or as the cans go in the door if a post loading check is not feasible) to make sure that the load at least matches the trim sheet. From a number of years on freighters, I have to say I didn't see gross errors often, but certainly saw them occasionally. Reload only means a delayed departure.
(b) having the certifying crew member (normally the captain) double check the numeric additions while a second crew member double checks the trim calculations using a load adjuster (slide rule style IU calculator). Takes negligible time and I have seen it save the day once or twice. This can be used for freighters or pax operations and is a sound philosophy if the load sheet is presented as a computer printout .. this latter, very common, technique is not usually checking friendly for the crew.
(c) on-board oleo check systems. These are rarely particularly accurate but those of which I have had experience generally have been consistent so a deviation in expected delta (error, if you like) is the alert signal.
.. and I certainly agree that some pilots, not seeing what goes on behind the scenes, can be guilty of apathy. Most, however, on larger aircraft (at least in my limited experience) have been attentive to the reasonable double checks which can be made ..
Are you able to make a case for real world accuracy and repeatability using a weigh bridge check system ? Specifically, one needs to address, satisfactorily, those sorts of error sources which Genghis, among others, has suggested as being worthy of concern ...