The utility of formal proofs implicitly assume that the formal system is consistent (except those weird paraconsistent logics). Proof by contradiction, and more generally resolution proofs, are explicitly propositional logic, which is consistent and complete. Incompleteness is introduced by rules of inference for manipulating quantifiers powerful enough to represent arithmetic, which is beyond mere proof by contradiction.
The incompleteness theorems tells us that formal systems that are powerful enough to represent Robinson arithmetic (or its stronger, more orthodox cousin Peano arithmetic) have undecidable statements, but propositional calculus isn't that powerful and is entirely decidable. Even Presburger arithmetic and extensions of it that involve multiplication by constants are SMT decidable, and are provably complete. So in short, resolution proofs themselves aren't handicapped by incompleteness.
Incompleteness means that its possible there's some new Russel's paradox out there that forces everyone to rebuild the foundations of systems strong enough to represent Robinson arithmetic. The consequences of such a hidden inconsistency is catastrophic, because then all statements can be proven (by contradiction, as you demonstrated) and the system becomes useless; But not necessarily irreparable, as we did recover from Russel's paradox. However, we can take comfort that for many useful weaker systems, they are on perfectly stable, decidable, complete ground.