However, several others are off by more than the statistical significance even in the hand-counted districts, including Clinton.

Choosing pairs of two likesized precincts in the same county, you get pairs where the demograpic differences should not be that large.

That's a definitely false assumption. Almost all counties include urban, suburban, smalltown and rural precints.

As for the linear fits: the odds of the result are less interesting than the odds of the  linear part of the result, and they don't even give an R-squared. (It seems to me a superposition of a ringing-off sinusoid and a linear would be amuch better fit for Clinton.) I should again re-state the point that there is no control for systematic errors.

_{*Lunatic*, n.
One whose delusions are out of fashion.}