Why do we need formal semantics for predicate logic?
A minor comment, and a more serious answer. 

First, it doesn't make sense to declare a natural deduction system complete by fiat. Natural deduction is interesting precisely because it has a natural internal notion of consistency and/or completeness -- namely, cut-elimination. This is a fantastic theorem, and IMO fully justifies attempts to give purely proof-theoretic semantics (and by the CH correspondence, it likewise justifies the use of operational methods in programming language semantics). But this is interesting precisely because it offers a more refined notion of getting the logic right than consistency. Taking the proof theoretic road means you'll have to do more work, but in exchange you get stronger results.

However, it happens that sometimes the logic per se takes a secondary role. We may begin with a (family of) models, and then look for ways of talking syntactically about them, using a logic. The soundness and completeness of a logic with respect to a family of models indicates that the logic really does capture everything both interesting and true you can say about the class of models. A concrete example of when models are more interesting than the logical theories occurs in program analysis and model checking. There, the usual thing to do is to take your model to be the execution of a program, and the logic to be some fragment of temporal logic. The propositions you can say in these languages are (deliberately) not terribly exciting -- e.g., null pointer dereferences never occur -- but it's the fact that it applies to actual program runs that gives it interest.