Yeah, they should not have lead with subterfuge. It's still remarkable to many people (myself included) that a pool as small as 23 gives a 50% probability.

I think even given that premise, the "50% probability" is still a bit of a rug pull. The casual listener still believes the problem should address the 100% match.

A more honest approach is to plainly ask how many people have to be at a party to guarantee there are at least two people with the same birthday. To even the layman, the answer is 366 of course. Follow that though with, "And how many people will have had to arrive for there to be a 50% likelihood that two people at the party have the same birthday?"

To go from 366 to 23 I think is a surprise to many people. Because humans suck at probability, most people might instinctively assume half of 366 (183). So it becomes a surprise how low (less than two dozen!) it really is.

My own "drunk walk" to making sense of the small number: when two people are at the party, it is intuitive to me that there is 1 in 365 chance they will have the same birthday. As soon as a 3rd person arrives though there are two partygoers they might match so the odds have just doubled! :-) I understand though that the 4th person arriving does not double the odds but nonetheless increase the chances by 50%.

Suddenly I can now see a kind of asymptotic curve that, when we get to 366, will at last cross the threshold for 100% probability. But the asymptotic nature makes it clear to me that it will cross the 50% mark much sooner than would a linear growth. I am already convinced at this point that your 23 number is probably a pretty good one.