Dismissing Curry–Howard without addressing its foundational and extricable relevance to computation and logic isn’t a rebuttal.
I am not saying that types cannot do this work. I am saying that to do this work you don't need types, and AL is the proof for that. Well, first-order logic is already the proof for that, but it doesn't have general binders.
Now, you are saying, whenever this work is done, it is Curry-Howard, but that is just plain wrong. Curry-Howard has a specific meaning, maybe read up on it.
You don’t get to do type-like work, then deny the structural analogy just because you renamed the machinery. It’s a type system built while insisting type systems are obsolete.