Statistically the # of ranks matter. I am a numbers person. I choose to trust the statistics from the NRMP. Each person can have their on opinion about it.
Although there are many variables that go into who matches, only 36% of the 146 US seniors who ranked 5 programs or less last year matched. Conversely, 64% failed to match. I would argue that is statistically relevant. I interpret this to mean: To increase your chances of matching, do all you can to receive as many interview invites as possible, interview as best you can at 11 - 16 programs, and rank all programs that you could live with matching at.
I agree with you that doing more interviews can make you either:
A) Jaded - feeling you're an "absolute lock" after hearing all the, "what can we do to get you here", combined with post-interview calls and e-mails saying your great and ranked to match, ranked highly or whatever. Therefore you decline to contact programs, play the game etc.

Exhausted - I think its tough for applicants to keep up their true level of excitement, positive body language, and game face on their 15th interview. Especially if it's the 4th interview of a mini-tour long weekend that went: social night/interview/social night/interview/FLIGHT/interview/FLIGHT/social night/interview.
- I agree with you that at some point, killing yourself to do 20 interviews can produce some diminishing returns.
- Clearly, it does only take one program to match. And it only takes one program to "like" you, or in other words, rank you high enough to match. However, I would approach the process in a manner that gives me and my paper application the best probability of matching. That means interviewing well and often and it means playing the interview and post-interview game well, and it means ranking as many programs as you can live with.
- I don't like any situation that yields a 36% success outcome. If I did find myself in that situation, with 5 or less programs on my ROL, I would certainly do anything humanly possible to try and move myself up the list of those programs. Sometimes your efforts will be futile and wont make a difference. Sometimes they might help. Who knows.
I think you nailed it on the head. The game needs to be played to a degree and certainly played well.
Good luck to all this year in the match!