the longhand calculation's integrity is maintained by whatever alternative loading calculations might be proposed
If we look at a fairly basic calculation, where the loading arms do not alter during flight, then the trim sheet should produce the same result as the longhand calculation for all intents and purposes. This, certainly, would be the case for, say, a small trainer where the occupants have no likely means of moving during flight.
However, as the aircraft gets a bit bigger, things start to become a little more complex. Take, for instance, the case of passengers moving to and from the WC or along an aisle for whatever purpose.
Were the calculations to be done by the standard longhand method, this would need multiple calculations to cover the CG variations in flight. Such a practice would become rather unwieldy. To cover this increasing complexity, the practical way to address in flight CG variation is to run some appropriate sums and then constrain the CG limits to ensure that, while the calculated CG might come with an error, the actual aircraft CG is maintained within the AFM/POH limits as prescribed. This is the basic thrust of what curtailment (the favoured US term) means. Whether the sums are done by longhand or trim sheet, the result will be the same with the same error associated with whatever problem exists. The same workaround (curtailment of the envelope) is what gets us out of the problems.
Generally, this process creates no problems. The only concern relates to using the calculated CG to set the takeoff stab trim for jets but, even then, a small variation in stab trim setting isn't going to create a significant concern.
The adjusted weight method is a quite clever technique developed by Peter Saunders, a Boeing weights engineer, in the 1960s. At that stage, the Industry was faced with the introduction of short haul jet operations. One of the problems with short haul, compared to international long haul, is that the turnarounds are far more rapid .. aircraft make money flying, not sitting on the ground.
So now, the weights folk in the airlines were faced with having only a few minutes, rather than, perhaps, an hour, to run the loading calculations. Keep in mind that electronic computers in general use was still some years away, so the existing long hand calculations (generally run with mechanical adding machines) became a major bottleneck.
During this period a great deal of development in loading system techniques occurred. However, given that many people are resistant to change, Boeing received a number of requests for a simple way to use the old techniques in a manner designed to get around the time problem.
Saunders, off his own bat, came up with his adjusted weight method (which upset the apple cart somewhat at Boeing as that was not the done thing .. another story for another time, perhaps).
The technique replaced the normal weight addition by the addition of a concatenated set of "numbers" where the weights were rounded to the nearer 100 and the final two digits became a CG-related value. If the system were set up carefully, the addition of a set of numbers could provide the loaded weight, CG, and stab trim setting in the one calculation. The method was adopted by a number of airlines and, once folk were trained and practised in the procedures, the sums could be run very quickly indeed.
Once the general introduction of mass-produced computers arrived in the next decade, the technique rapidly became a bit of a dinosaur process and was replaced by using computers to do the standard number crunching.
So far as error is concerned, the technique introduced a deliberate error. However, provided the aircraft was reasonably large, the magnitude of the error got lost in the general calculation noise and was addressed by the usual curtailment methodology. Saunders' assessment was that the technique was not appropriate to aircraft with MLW less than around 80,000 lb (ie around 35-40 tonnes) providing that certain constraints were observed in developing the technique for a given aircraft Type/Model.
I understand that a few operators still use the technique but, in essence, it is dead and buried. I am not aware that it was ever used in Australia.
You can presume not to come across it in action and, in any case, it would not be seen in smaller aircraft so not a technique with any relevance to basic pilot training.