This scrutinization ignores the randomness factor. Some people got stuck in traffic or maybe had some emergency that prevented them from voting. Some people went to the wrong voting area or their registration got lost. Some people just plain filled in the wrong oval, voting for the wrong candidate. These problems are rare and roughly cancel themselves out but in a very close election like Minnesota the randomness greatly outweighs the disputed ballots. Yet because of the hard cut-off the scrutinized ballots get the most attention.
I suggest we eliminate the hard-cut off by adding our own randomness. For simplicity suppose we had two candidates, Al and Norm, who received x and y votes respectively. We flip z=x+y uniform random coins. If the number of heads is at most x we declare Al the winner and Norm the winner otherwise. If x > z/2+ 10 z1/2 then this process would make Al the winner almost all the time. But as x and y get very close the probability that Al wins approaches 1/2. Since a change of a few disputed votes won't affect the probability dramatically, we won't have so many battles over so little.