That logic is crazy. You say the winner needs to have a lot of second choice votes (as well as first choice otherwise they would have been eliminated) then say that the one with the most third choices would win - you aren't making your conclusion on logic, you are trying to make the facts sound like they point to the conclusion when they don't - a bit like saying Man has never been to the mood because tin-tin was a cartoon. Its just idiotic
If you use the school analogy better to say 'where will the funds go to' (as in the school analogy all can't go to the preferred school).
First past the post works like this -
46% of people vote for the money to go to school one - the school their kids are at.
School 2 has 30% of the kids in the area in attendance, but a third argue that the money should go to the school with most need and put this as second preference.
School 3 has 24% of the pupils and think the money should go to them - the most disadvantaged school but if they don't get it, school 2 should get it because the best school is already the best school and they feel one of the weaker schools should get the money. They also know that a third of the people in that school support their development. They all put 2 as second preference.
First past the post - the money repeatedly goes to school one, who remains the best school because of that, even thought this double- disadvantages 54% of the population -kids in the best school think this is fair because 'everyone voted for it'.
AV - 46% of people think the money should go to school one in round 1. They lose round 1 because there are more opinions in opposition to that. Round 2 kicks out school 3. all school 3's second preferences go to school 2 and 2 wins.
School 1 kids are unhappy - they always win so they hate the new system. School 2 kids are happy - they get the money, - a third of them still support developing school 3 and elect to share some of their facilities because they know that school 3 is happy to support their aims if they can't benefit directly themselves.
School 3 kids are less happy than getting the money, but more happy than all the money going repeatedly to the same school. 54% of the constituency are more happy with the new arrangement than with the old biased in favour of the largest interest group, even if they don't get their absolute preferred outcome.