So who played this wrong?

Preflop:
In a midstakes tournament your open raising size would be perfectly fine. In Freerolls and low limit tournaments you should open raise 3BB until the blinds get a big bigger

Flop:
As a preflop aggressor you should always contibet boards like this. You can perfectly represent Ax and therefore you will win the pot quite often. It doesn't make sense to check behind since you don't have a drawing type of hand and won't be able to call his bets without improving.

Turn:
Well, the chance that he has Ax gets smaller. The card is a good one for you since it is more likely that you are in front. His small bet indicates weakness (at least in most cases) as he tries to take a step at the pot without paying many chips. You can call or raise in this situation. Imo a raise would be the better solution for 3 reasons:
1) You have the opportunity to win the pot immediately.
2) You will gain further information on your opponents hand (if he 3bets you can easily fold)
3) You will get the initiative back - since you are in position you can choose to valuebet the river or check behind in most cases since your opponent will check the river after your raise (if he calls)

River:
You hit your set. You shouldn't be afraid of better holdings - there is no indicator that your opponent might have a better hand. Your Value-Raise is way to small imo. At the same time a shove would be way too strong. I would Raise to ~800.
As played: No way of getting out at this point. You have to bring your stack in
There are a lot of worse hands that might create action here. Many Ax hands, busted Flushdraws, maybe hands like T8 or something or just random bluffs.

Your conclusion is right btw

Regards,
Asaban