It should be apparent to e.g. anyone running a manufacturing business in which certain procedures need to be followed to avoid delivering defective stuff to customers that the written log method ... is wide open to gross errors.
Well, we know that, which is why most quality systems don't rely on everything being recorded only once; when I'm looking for the historical background to a cock-up, which as a software engineer I find myself doing from time to time, it is helpful if the quality system has more than one audit trail -
three is nice, then you normally have a reasonable chance of working out which record is wrong, and what actually happened, and who did it.
Here one might hope there would indeed be three records - the log of what people thought was in the plane, as referred to, and the fuel pump log as to what was put into which plane (this must exist because the fuel operator needs it for billing, and billing is important), plus a record of flying hours possibly verifiable by some sort of meter in the airfcraft (this must exist because the airfcraft operator nees it for billing, and billing is important). The log of how much fuel was in the plane at the start of each flight should be reconstructable from the other two records.
Sure, it's unreasonable to expect the pilot to demand to see and check the fuel operator's invoice book, so he wouldn't have access to all three sources. That doesn't explain why the CAA couldn't reconstruct it though.