Extremely narrow election outcomes—such as could be reversed by rejecting a few thousand ballots—are likely to trigger dispute over the results. Narrow vote tallies may generate recounts and litigation; they may be resolved by courts or elections administrators (e.g., Secretaries of State disqualifying ballots) rather than by voters; and they may reduce the peacefulness, perceived legitimacy, or predictability of the transfer of political power.
This in itself shows some of the problem with the argument, since this is an immensely naive statement. In a large election system, any election outcome is likely to trigger dispute over the results under a wide variety of conditions, not just extremely narrow ones. We have recounts and litigation in every presidential election before the Electoral College meets; most gubernatorial races have recounts and litigations. None of this need have anything to do with the outcome being 'narrow'. It happens when there's suspicion of voter fraud. It happens when there's suspicion of voter suppression. (And in the United States, we have suspicion of fraud and suppression in every single election year, even independently of cases in which either is well-founded, because Republicans use voter fraud suspicions and Democrats use voter suppression suspicions in every single election they lose in the attempt to blunt the force of having lost and in order to raise funds for future elections.) Election administrators are making decisions about what ballots to disqualify in every single election, because disqualification is not a matter decided by voters but by the policies established by the legislatures, the executives, and the courts. In California, it's hard to disqualify ballots; in some other states, all sorts of things can disqualify a ballot; this is not because of the voters, but entirely because no state has exactly the same mix of aims in structuring how votes are counted, and that mix is determined entirely by a complicated negotiation among the relevant authorities. You can't have an election without counting ballots; counting ballots depends on the methods for counting and discounting; in no election system is this established directly by the voters, because that would give us an infinite regress. Election disputes very rarely arise from the method of election; they arise as a regular occurrence from the politics of elections. The "peacefulness, perceived legitimacy, or predictability of the transfer of political power" is not a byproduct of counting practices.
That aside, it's not very clear why one should care about whether elections are extremely narrow or not. While our usual way of describing this -- that the election was "decided" by 20000 votes or fewer -- makes it sound like that it's a few people deciding the matter, in reality, this is a figure of speech. What wins the election is all the votes, not just the surplus of the win. And while I don't myself believe in popular mandates, lots of people do, so that extremely narrow wins in controversial elections serve the function of limiting the plausibility of a claim of having a clear mandate from the voters.