It's a little more impactful than that. Have a look at Z3, Boogie, Dafny, and similar technologies to see practical application of what I mean. It boils down to there being "non-general" algorithms that still work for virtually every input you're ever going to give them. A hypothetical algorithm that decides the halting problem for 99.9999999% of programs would not violate the impossibility proof.

The limit case is maybe interesting. What about the algorithm that decides the halting problem for every program except for one? Does the impossibility proof prohibit such an algorithm? Does it make a difference if the identity of the unique program is known or unknown?

And then of course there is classic pen and paper hand derivation like the old guard (Knuth and his peers) did. The claim that that is following an algorithm is yet to be proved or disproved.