No, they aren't -- they are all Turing-complete, look it up. The THP applies to all non-trivial programs, meaning programs that support Peano arithmetic or anything more involved. That limitation certainly applies to all the programs you're describing.
> Pop-sci explanations of the halting problem have probably done incredible damage to formal verification as a whole.
That's pretty funny. It's not every day I hear Alan Turing and Kurt Gödel described as purveyors of "pop-sci".
> Mostly because it makes people with only the most cursory understanding of it go "why bother, it won't work anyway."
I won't try to list all the logical fallacies in your argument. You could do that if you were so inclined.
That means we can't make the claim that a particular program, running in that language, is error-free. I don't get why people find this so hard to understand.
> That would imply you could never know anything about a program.
No, it means we can't make the claim that a program is error-free. That's the simplest statement of the Halting Problem.
When someone applies Lean to verify the correctness of a chain of logic, they must remember than Lean cannot be proven to be error-free, because ... wait for it .. Lean is Turing-complete.
But most real-world programs are maybe not that "expressive" since we don't even want them to be capable of infinite loops (since those would often be denial of service vectors). So you should usually be able to prove that they halt (and presumably other properties as well), especially when you allow agents to rewrite things to pull out invariants that make proofs easier (as they did).
And for cases where getting clean invariants are difficult, I imagine you can just hack it by having some cutoff which trivially ensures things always halt. (Not sure what the equivalent would be for "no bug" proofs, probably rejecting things that are technically valid input per spec but seem unusual?)
I would say, in practice, you are more likely to encounter complexity barrier (we don't know how to decide some NP-hard problem fast enough) to proving things about your program than you're to encounter a true undecidability barrier (Rice theorem).
Although the prover might be smart enough to say, "this function is getting kinda complicated, let me model it as a random variable and see if it's enough". I think LLMs can kinda do it but not sure if your typical theorem provers can (it's on the level of being able to employ probabilistic proof techniques).
Pop-sci explanations of the halting problem have probably done incredible damage to formal verification as a whole. Mostly because it makes people with only the most cursory understanding of it go "why bother, it won't work anyway."