Just because a system can encode paradoxes doesn't mean the system itself can't be solved. Turing, Chaitin, Godel didn't prove things can't be solved, they actually contributed by solving specific areas of specific systems.

People didn't give up on ZFC because of Godel's theorem. There's currently 42 or 43 unproven 5-state Turing machines we need to confirm BB(5).

We are solving the halting problem. We are solving ZFC. That's the entire point of mathematics.

Turing's proof of the halting problem being undecidable is extraordinarily vague. It shows that for any halting procedure, there is a program (without any constraints) which violate its output. It does not, for example, rule out whether you can write a Halt procedure for no-input Turing Machines with N states. In fact we have already written them for N=1,2,3,4.

Considering Chaitin-Kolmogorov complexity, of course we can't use N bits to describe unbounded information. This doesn't in any way preclude us from making an N bit program to describe halting behavior of K<N bit programs for some N.