As I read it, the action is "You may: [close 1 and draw a card]. Then, you may: [strike]." Technically the first clause is optional, but at that point it's just the same as taking the prepare or strike actions. (I'm pretty sure you don't get to draw and strike without closing, but correct me if I'm wrong.)
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"As an action, you may [do something]" is just how actions are templated. I don't think it's really saying "Action: You may [do something]", but rather "You may do this as an action: [do something]". It reads naturally, which makes the game more approachable.
You can see it used on the characters abilities of Umina, Ryu, Bang, etc., when there's really no reason for their effects to be optional. D'janette doesn't use it, but that's a compromise because it literally won't fit in her text box. S4 is inconsistent with it, but S4 is kind of notorious for bad templating. Newer characters obviously use "Action:", which is shorter and removes this ambiguity.
But I think it's fair to say that "close 1 and draw 1" is a mandatory effect. Even if it weren't, it's both or neither, because it's "you may [close 1 and draw 1]", so there's no reason to take that action if you're not going to do it (otherwise, Strike or Change Cards 0). At the very least, it's ambiguous, but it is clearly against the design ethos of Ken to have such a fiddly effect, and it's especially problematic in a digital implementation to have pointlessly optional effects, since every extra decision requires input and wastes time.