If there is a proven theorem that something is impossible, either the theorem is wrong or it's impossible.
> the halting problem is solved routinely by people in software engineering research groups
I'm not sure how I'm meant to read this.
If it is that people do the solving of the halting problem (that is, people read the source of the program and say "this will halt" or "this will not halt"), then they have not solved the halting problem: the halting problem states there is no way to do this by algorithm.
If it is that people write software which solves the halting problem, then you are wrong, and you explain to yourself why you are wrong. They are solving a different problem, namely, whether some very small class of computable functions halts.
Your two sentences "the halting problem is solved routinely" (assuming you mean the latter case, as above) and "this does not invalidate the Halting Theorem" do not logically work.