A digression: But hopefully at this point, you'll realize the difference between the frequentist and Bayesian instincts in this situation. [...]

Yep. This really is a digression which is why I hadn't brought up another interesting example with the same group of friends:

One of my friends dealt hearts in a manner of giving each player a pack of three cards, the next player a pack of three cards and so on. The amount of cards being dealt were the same but we all complained that this actually affected the game because shuffling isn't truly random and it was mucking with the odds.

We didn't do any tests on the subject because we really just wanted the annoying kid to stop dealing weird. But, now that I think about it, it should be relatively easy to test...

Also related, I have learned a few magic tricks in my time. I understand that shuffling is a tricksy business. Plenty of more amusing stories are lurking about. This one is marginally related:

At a poker game with friends of friends there was one player who shuffled by cutting the cards. No riffles, no complicated cuts, just take a chunk from the top and put it on the bottom. Me and the mathematician friend from my first example told him to knock it off and shuffle the cards. He tried to convince us he was randomizing the deck. We told him to knock it off and shuffle the cards. He obliged while claiming that it really doesn't matter.

This example is a counterpoint to the original. Here is someone claiming that it doesn't matter when the math says it most certainly does. The aforementioned cheater-heuristic would have prevented this player from doing something Bad. I honestly have no idea if he was just lying to us or was completely clueless but I couldn't help but be extremely suspicious when he ended up winning first place later that night.