That "proof" is applicable to any algorithm that "exists". As "an algorithm exists, so we can enumerate all of the Turing machines "in parallel" and find it" would work for anything in P, other algorithms as well. However, good luck actually running that algorithm... Enumerating Turing machines in parallel, executing them, and n could be many times larger than the age of the universe. You want something that you can execute, not something that in theory exists.
The kind of ultra-shoddy "construction" you get from "Here is an impossible to build machine that would give us the proof" would still not get you a demonstration. It would still be something we don't have the algorithm for.