This is a very interesting philosophical question! There are philosophers on both sides of the issue. A prominent figure who spoke against SOL as logic was W.V.O. Quine. His argument mostly focuses on the point that SOL, which quantifies over sets explicitly, seems to just be "set theory in disguise", and hence not logic proper (see his Philosophy of Logic for more details). People have also put forth the argument you presented, viz. any putative logic must have a sound and complete deduction system in order to be a genuine logic, and since SOL doesn't have one, it follows that it can't be a logic.
On the other side, a prominent defender of SOL as logic proper is George Boolos. See his "On Second-Order Logic" and his "To Be is to Be the Value of a Variable (Or to Be Some Values of Some Variables)". One argument put forward in favor of SOL from a practical standpoint is that it has a way of really shortening proofs (see "A Curious Inference"). Boolos also suggests that we could use monadic SOL to model talk of plurals, i.e. to model sentences like "Some critics only admire one another". Another great defender of SOL is Stewart Shapiro, whose book Foundations Without Foundationalism not only presents a wonderful technical treatment of SOL but also some good philosophical defenses of it. In particular, Shapiro argues that SOL is needed for everyday mathematical discourse. I would highly recommend this book as an introduction to the subject, both mathematically and philosophically.
The status of completeness theorems in this debate is fascinating, but difficult to pin down. After all, many philosophers might just say "Look, so what if SOL is incomplete? That just means that the set of validities doesn't coincide with the set of provable sentences, i.e. there are some sentences which are logically true, but not provable. And while this is unfortunate, why shouldn't we expect this? Why shouldn't we just accept that logic didn't turn out the way we hoped?" On the other hand, many philosophers might say "Look, logic is about making inferences. If there are truths which don't come from some deduction or inference, then how could they be truths of logic?"
I personally don't find the view that genuine logics must have complete proof systems convincing (or at least convincing enough), but I certainly do feel the pull to search for/work with logics with complete proof systems. Ultimately, it comes down to what you think "logic" and "logical consequence" are, which are highly contentious matters in the philosophical literature, and which have no widely accepted answers.