> However, my understanding is that the incompleteness results apply to only recursively enumerable axiomatic systems. I can find references for this.

That's a matter of semantics as to what you consider the first incompleteness theorem to be precisely (of which there are several variants). Gödel's proof itself doesn't directly work for second-order logic. But the statement "if Γ is some axiomatic system that satisfies certain conditions, then for any sound proof calculus there is a sentence that isn't provable from Γ in this calculus" is true in second order logic too, it's just that the "failure" happens much "earlier" (and is in some sense obvious) than in the case of FOL.

> PA_2 is not such a system and as such the Incompleteness Theorems don’t apply.

I'm really not all that familiar with second-order PA, but it is my understanding that the set of its axioms is decidable. It consists of a finite collection of axioms plus one schema (comprehension axiom) which is valid when it's instantiated by any given sentence - but deciding whether something is a valid sentence is easy. Therefore, what you quoted applies to second-order PA too.