The one thing missing from that video (at least up to 4:23 when I got frustrated - and he had explicitly disclaimed talking about the Pauli Exclusion Principle before this point) which gets really to the heart of it is that the Pauli Exclusion Principle kicks in when one thing literally runs into the other - when parts of two things were trying to occupy exactly the same state. If 'couldn't go any further or you'd be inside the other thing, but you can't do that' isn't 'contact' then the word has no meaning.

The interviewer is exactly right at 4:17 - he did the demonstration wrong. He should have brought them into contact. Only when he was pushing inwards and the balls were pushing back hard enough to balance -- that's when he'd say they're in contact.

So this isn't a great example because the proper explanation does include a smaller version of what's being explained.