Proving a language as undecidable without using reductions
It isn't recognizable because it is infinite, therefore it'll loop infinitely. Does this work?


No, because that claim is false. And your writing isn't at all clear, here. Recognizability is a property of languages, so your first two "it"s must refer to languages; but "loops infinitely" is a property of Turing machines, so your third "it" must be one of those.

Ultimately, though, your whole approach is doomed because the languages you call $One$ and $\overline{One}$ are both decidable. You can easily see that $One$ is decidabe just by designing a Turing machine that decides it.