Erina has never been the antagonist, she's been the rival of sorts, she's never done anything to antagonise them, her actions were down to her conflicted feeling, where Souma's techniques clashed with her teachings. even the Kuga battle, you couldn't consider him the Antagonist as it was Souma that sought him out and challenged him, Kuma was a competition, these two would not have been considered antagonists even if they had joined the Elite 10 (well we would likely see a personality shift from Erina as she'd likely be emotionally cut off under the control of her father.)
Even now the rest of the elite 10 seem to behave like competitors than actual antagonists, so he just sticks out like a sore thumb, this is the fault of the manga by design. Concentrating on Eizen would be fine if not for him already being shown to have been beaten at his own game, the Karage arc, the dorm closure arc, these are all defeats by him, and lets not forget Eizen is the weakest member of the Elite members, and even if he is the only one that behaves like a antagonist, he's still the weakest link (after the throw away members in the first round).
We've already been given details of his underhanded techniques, we had it pointed how he is known as the alchemist in both his previous defeats, so there was really nothing new to bring to the table, if anything this should have been a perfect opportunity to show him use his alchemist ability to defeat Takumi on talent alone instead of using underhanded techniques, we could see clearly his defeat to Souma affected him, so it would be fair to see him go beyond his usual tactics to secure a win, but instead the mangaka chose to have him stand still.
This was done to so show Takumi has overcome his previous lost where he was baited into making a error and walking into the trap the copy-cat had set for him...instead he took what he leart in that loss and baited Eizen into walking into the trap he had set. I guess the moral about this match is learning from your losses, Eizen didn't where Takumi did.
She was the closest thing we had to a main antagonist other then the school itself (which she is a representative of) until Azami came.
She was an antagonist is as much as she wanted Souma out of the school, and she represented "the elitism problem that the school has." Which is also why she rarely tried anything to ever try to get him thrown out. Since she figure that he was so unworthy of being in the school; that he would be thrown out on his own because of his own worthlessness. Then Souma surprised everyone when he was able to surpasses expectations.
She wasn't much of a rival at the time because two reason; her skills were so beyond the rest of the cast -when Akira won the Autumn election he said he was now the second best first year behind Erina-, and Souma's goal of making her admit his food was good made her a goal. Now that she is on the main cast side she is now; a rival, a love interest, the main heroine, and one of Souma's goals.
To be honest the Elite ten if this was another manga/anime would have been the main protagonists if this was an anime of all powerful student council in a cooking school. As for Eizen, the thing is that each of those previous clashes were not him going all out in a cooking battle; the first one was against his business skills, then by using one of his minions, and then against his attempt to make someone buckle under outside pressure. So this is the "final defeat of a recurring villain." Again he is the closest member of the group that comes of as completely dis-likable, and who the audience wants to see defeated.
And your correct about the Takumi getting revenge for his previous defeats, and the moral.
I feel so in a sense
