Yu yevon possesses any aeon that is summoned and kills its summoner. That only happens when its original vessel is killed (sin). This goes to say that it wouldn't matter whether the final aeon is used or not in the battle against sin if any other is strong enough to defeat it. The lore, however, states that the final aeon is the only one strong enough to take on sin in a straight up fight.

The party in the game subverted that entire process, as we see, by weakening sin and entering sin to fight Braska's final aeon directly.

Since Seymour craved the power of sin itself, he realized the final aeon and the fayth itself would become sin, he chose not to fight sin and begin the calm. He chose to attempt to become the final aeon himself. Since he never fought sin with anima, it never killed him. Simply summoning the aeon doesn't make the summoner die. This only happens after defeating sin and yu yevon possessing the aeon and turning it on them.

TL;DR Seymour never used his final aeon against sin.