I follow your logic - to a certain extent - but I don’t see how one uses statistics to conduct a recount. What you say may convince a judge that a recount is needed but how does one actually compute the new results? It would seem that would have to be done one vote at a time.
It's not a recount, it's invalidating the outcome with a certain degree of certainty. Recount won't work here, because some of the ballots, and we don't know which ones, were cast illegally. If the number of illegal ballots is lower than the margin of victory, then the illegal ballots don;t change the outcome, even if all of them were cast for the winner.
And you never get any precise result, like you do with counting. You get an estimated count with a margin of error and a confidence level. At some margin of error and confidence level, you'd be justified in changing the outcome. I gave a hypothetical example above. Margin of victory is 2,000; contest was roughly 50-50. There are 5,000 invalid ballots. I can say with very very high confidence that the number of ballots for the winner is between 2,400 and 2,600; and with even greater confidence, say the number for the winner is more than the margin of victory. I can't tell you an exact number - like you'd get in a count. But I can be damn certain that the contest was called incorrectly.
The counterargument is that the invalid ballots went primary to the loser - that is, that invalid ballots aren't split between winner and loser the same way the total/outcome is. The winner argues that all the invalid ballots went to the loser.