Not only is there nothing here that approximates a "solution" to an unsolvable problem, but that there are more programs you can decide the more time you simulate them is the actual statement of generalised halting theorems. Indeed, this is literally how they are summarised: "Informally, these theorems say that given more time, a Turing machine can solve more problems." [1]

When someone says that you can, in practice, sort-of get around the halting theorem by noting you can solve more problems given more time, do you see how it's not an approximation of a "solution" that gets around any theorem but the very point of the theorems? If you see you can solve more things in more time you are observing the halting theorem and its generalisations at work, not getting around them in practice.

But that's not the end of things because now we can ask what it means to "get around" or "approximately get around" a theorem that basically says that you can buy a bike for $50 but you can't buy a car for that price. Saying, "I don't mind riding a bike" is clearly not getting around anything, but saying "most people wouldn't mind riding a bike" could, indeed, be a way to state that the limitations of the theorem don't affect many people in practice. But such a statement also isn't true.

If someone wants to claim that in practice we rarely come across intractability where it matters, that would, indeed, be interesting, but I also don't think it's true. So if someone wants to say that most problems that are interesting in practice are decidable, that may well be true, but even Turing realised that decidability isn't a very interesting property. Decidability was replaced by tractability in the 70s, as halting was replaced by the more general and precise time hierarchy in 1965. We more precisely discovered that not only are there problems that cannot be solved in any time bound, but that there are problems that can be solved in 2^N that cannot be solved in N steps.

And if someone wants to say that we don't frequently come across intractable problems in practice, then that is not true.

If anything, the limits we run across in practice are more constrained than the known theoretical ones. Even though we have yet to prove a hierarchy between P and NP, and even between P and PSPACE, we do find it much harder, in practice, to solve problems that are complete in NP than problems known to be in P, and problems that are complete in PSPACE are harder in practice than those that are complete in NP.

[1]: https://en.wikipedia.org/wiki/Time_hierarchy_theorem

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."