Excellent point. This misconception is also the root of silly statements like "human intelligence is uncomputable because we can do things like tell whether a program will halt." No, we can solve specific cases of the halting problem, we can't solve the general case any more than a computer can.
> we can't solve the general case any more than a computer can
While almost certainly the case, I'm not sure how we can demonstrate that this is true in principle without begging the question, if we're asking whether human intelligence is computable.
I agree that all we have evidence of is that we can solve specific cases, and that this completely undermines the argument you're critiquing.
This error also occurs more subtly. I had someone tell me that static type checking was basically useless because it couldn't even do something as simple as saying whether a program will halt. Nevermind, if we're demanding this of all programs, that neither can any other method of reasoning we're applying. (And also nevermind, if we're demanding this of an interesting subset of programs, that there are type systems that will tell us exactly that...).
