One the one hand, companies like Amazon whine (to whichever politician is listening) that there's a desperate shortage of skilled technical people. On the other hand, they subject new hires to 14 interviews.
Tell me: is this the sort of thing you do when you're in desperate need of employees?
Let's assume that your interviewers are uniformly exceptionally good, and have a false-negative rate of only 10% (i.e. 90% of all good applicants are accepted).
An interview with a good candidate is therefore a Bernoulli trial, and the outcome of an interview loop can be modeled exactly as a binomial process. If you conduct 5 interviews, the probability of at least one interviewer saying no to a good candidate is 0.41. If you conduct fourteen interviews, the probability of at least one false negative is 0.77, and the probability of at least two false negatives is 0.42. In other words...you've made it twice as likely that you're going to have someone falsely reject a good candidate.
Meanwhile, let's assume that these unrealistically brilliant interviewers are even better at rejecting bad applicants: they each have a false-positive rate of 5% (and therefore, a true negative rate of 95%). If we assume a five-interviewer loop, the chance of a single false positive is 0.23. But bumping the number of interviewers to fourteen raises the chance of a false-positive to 0.51! With fourteen interviewers, the chance that one will say "Hire" to a bad applicant is over 50%!
In other words, with every interviewer you add to a loop, you raise the chances of a false-positive or (even more likely) a false negative decision. If you have normal interviewers (i.e. people who don't have super-human discriminatory ability), the problem is even more pronounced.
But, you may ask, what about the case where they should be rejected, 8-6 (assuming majority voting not consensus, but you could apply this to any voting threshold), and you have two false positives and 0 false negatives, is it really that horrible that you might experience the borderline case where you hire someone who might be mediocre, assuming everyone else is pretty good?
Yeah, there is a good chance you'll makes some individual mistakes. But having lots of interviews decreases the probability of making those mistakes frequently enough to actually influence the outcome.
Hate to say it, but your "mathematical justification" is really just another example of geek chest-thumping.
Your second argument is more reasonable: if you stick to strict majority-rule voting, it's true that you're less likely to get a false-negative result with a larger number of voters. (You'll also increase the number of false-positive results). Problem is, nobody actually does it that way. Most hiring managers are trying to keep the false-positive rate low, and tending toward consensus. Thus, random false-negatives have quite a bit of impact -- one or two "no-hire" votes are enough to torpedo a candidate at places like Amazon, Microsoft and Google.
Clearly, when you do enough interviews, you start to expect "no-hire" votes, just due to chance. You can claim that this is good for the company, but like I said before...these companies do a lot of complaining that they can't find "qualified" applicants. Perhaps if they re-thought screening processes that involved dozens of interviews, they might find that the qualified applicants were there all along.
"Lowering standards" doesn't work as a sound bite, but if something is too high, one lower it, even if it is a standard. I don't think standards are sacred.
For instance, if you used to be a very desirable mate and had standards to match, but are now less desirable, you should lower your standards accordingly.