Evidently not. There's nothing more about it in the final draft script:
There's still no explanation for "Another Earth." I think Spies (pronounced "Spees") was just trying to be "Sci-Fiey."
And the Star Trek Compendium
's discussion of the first draft doesn't mention anything about the "other Earth" aspect. I think it was just a throwaway element inserted for two reasons: a) to provide an effective teaser to grab the audience, and b) to justify the use of location shooting at the Culver City backlot to stand in for an "alien" planet (which is a whole lot more affordable than building a ruined alien city would've been).
This was the '60s, after all, and when it came to stories about space in the mass media, there was a sense that anything was possible, yet at the same time mass audiences and TV writers didn't have much experience with the full range of SF imagination you'd find in literature. So sometimes you got stories where people went to an alien world and found it was just like home... except everyone was 70 feet tall! Or except they were all telepaths! Or except most of them were left-handed! Audiences back then would've accepted it.
I've always had the impression that Adrian Spies was feeling toward a parallel-timeline sort of story. Miri's Earth collapsed at a 20th-century level 300 years before the episode, and it's pretty much what Earth itself would be if it had fallen prey to the same kind of experiment-gone-wrong. But such ideas weren't yet clearly delineated in the mind of the SFTV viewer or television writer of the era, so instead of presenting an actual method for crossing timelines, Spies just went with a more brute-force kind of "alternate Earth," one that a spaceship could just stumble across.