Actually sampling error increases disproportionally when moving to smaller n and larger number of values, so there is a much bigger impact between 20 and 40 (definitely not negligible) than between 2000 and 4000 (especially if we add that randomness of the sampling is actually an untenable assumption here), and even bigger 10 and 20 and massive between 2 and 4.

For instance a full F cabin is 14 people and offers a choice of 5 mains. It is aa very small n already for 5 categorical values, so on 100%, there would almost certainly be some error, but airlines hedge against it by over-catering, I think at 130% in F. Now imagine 9 people pre-order so BA need to apply their estimate for 5 people. They will load 6 (120%) or 7 (140%) dishes so either 3 or 4 of the five options will be in a load of 1. Chances that things will go wrong will be high, in fact at that level, there is more than 50% chance at least one person will not get their first choice if we assume BA's model was absolutely right!!

In CW, there are more people and only 4 options but I think overcatering is at 110% for mains if I remember, probably less on night flights. Now again, on a 60+ cabin you'll be fine but when 40 or 50 of them pre-order the rest may well experience problems.

In fact, most FTers already know this (if not necessarily the mechanism that leads to it) because the vast majority of reported choice problems we get are on relatively empty cabins.