Since contradiction proofs are arguments that operate in connection with the structure of proofs in general rather than exclusively in terms of the specific problem, then the entire argument may collapse on a technicality like "very large" vs "infinite". It's almost like the truth status of the argument is sharply discontinuous: you can't modulate the argument slightly and arrive at sensibly related assertions.
Whereas if the argument were made solely in terms of the problem itself (without the added meta-layer argument from the contradiction proof), but still used the halting problem, then we might consider the fact that in reality our computers have finite state so are technically exempt, but the argument still holds for all intents and purposes because of how large the finite state is.
Or is there another reason these arguments are generally bogus?