You're using "Closed Sicilian" a bit loosely here, the Closed Sicilian is 2. Nc3, the Alapin and Delayed Alapin are not the Closed or Open, they are their own thing.
As a Classical player, I do play 2...Nc6 after 2. Nc3. I have that freedom because I intend to play Nc6 in the Open anyway, so I don't care if White plays Nf3 and d4 and transposes back into the Open. That's the nice thing about the Classical.
I don't see why you would rather have played Nc6 if White plays the Delayed Alapin. You just have to play Nf6 or d5 and either way you end up back in the boring Alapin main lines. In fact, in Plichta's Anti-Anti-Sicilians course on Chessable, which I own, he states that this line has no independent value because it always transposes into the Alapin, and simply refers the reader to the section on the Alapin. Against the Delayed Alapin after 2...d6, I get to play 3...Nf6 with useful pressure against e4 and White does not have 4. e5 available anymore, so the positions don't transpose into the Alapin, they retain the character of a Sicilian. I'm happy to see this from White because I don't think it's very good and I feel like I get good play.
I personally enjoy 2...Kc6 as opposed to 2...d6 when playing the Sicilian defense with the black pieces. Saving the tempo on the d pawn allows you to play d5 in one move if white chooses to play the closed Sicilian, that combined with the central control that the knight gives you helps you take the center from white much easier. If you are playing the Sicilian defense, then chances are that you want an aggressive game, otherwise you would have played the French or the Caro Kann with the black pieces against 1.e4. You can have an aggressive game whether you play 2....Kc6 or 2....d6 if white chooses to play the open Sicilian, but white can choose to play a delayed closed Sicilian after 2Nf3 (like the delayed Alapin for example). I remember freaking out when I used to play 2...d6 after my opponent goes into a closed Sicilian, because I remember hating going into a closed, boring, timid, and slow game, but after playing 2...Kc6 I always find anti Sicilians to be a joke (except for the Rossolimo) because I know that taking the center from white is not that difficult unless they are +2000. What do you guys think?