The halting problem can be approximated by a sequence of increasingly accurate computable functions - "partial halting oracles" which give the right answer on "many" inputs, with each better than the last.

The sequences "converge to" or "approximate increasingly well" the true halting function in that for any input, there is some point in the sequence such that all subsequent partial halting oracles analyze its behavior correctly.

The halting problem is "unsolvable" because the goalposts are very high. An algorithm to "decide" the halting problem can have no "failure modes" of any kind, even if the probability of failure is vanishingly small. It must work on every single program. As soon as you limit the scope of the programs you care about analyzing in any reasonable way, like "deterministic programs without randomness that use at most 32GB RAM," the proof no longer applies.

The complexity classes you refer to don't conflict with any of this. In the general case, it is undecidable to analyze what complexity class an algorithm (or decision problem) is in, for instance, but this isn't usually summarized as "computational complexity analysis is an unsolvable problem."