[mid50sbaseball]

well...after reviewing some rules as you suggested it appears this ENTIRE CONVERSATION may be moot, lol.

someone please feel free to correct me if I'm interpreting this incorrectly.

so the intial condition was that a runner reached 3rd because of a wild throw while he was attempting to steal 2nd.

MLB rule 9.12.a.5

An error is a statistic charged against a fielder whose action has
assisted the team on offense, as set forth in this Rule 9.12 (Rule
10.12).
(a) The official scorer shall charge an error against any fielder:
(5) whose wild throw permits a runner to reach a base safely,
when in the scorer’s judgment a good throw would have
put out the runner, unless such wild throw is made
attempting to prevent a stolen base;

so if I'm reading that correctly, no error was actually committed, thus the runner, having reached 3rd via the overthrow and subsequently scoring on the single would be considered an earned run.

again, feel free to correct me if I've misinterpreted something.