Can someone explain to me how I read it wrong?

I read it as 2 cousins are in an accident and 1 of the cousins is the son of the surgeon.

What was the contradictory statement that I missed?

You read it right. There's no contradiction. The famous original bit started with "a man and his son". This bit is certainly part of the LLM's training corpus, so it's expected to acknowledg it when you mention it.

The thing is, you didn't mention that bit to the LLM. You mentioned a completely different scenario, basically two persons who happen to be cousins. But you used the same style when presenting it. The issue is not a hidden contradiction or a riddle, the issue is that the LLM completely ignored the logical consequences of the scenario you presented.

It's like asking it about the name of the brave greek hero in the battle where the famous Trojan Cow was present. If you get "Achilles" is obviously wrong, there was never a Trojan Cow to begin with!

There isn't a contradiction. Making the LLM look for a nonexistent contradiction is the point of this prompt.