There's an obvious intuition that the author seems to be missing with the 'two aces' paradox.
If you have a hand with two aces, you have two chances of getting the "Ace of Spades." Having a hand with two aces doesn't make it any more likely to have "an ace" than having a hand with one ace does.
If you have a hand with two aces, you have two chances of getting the "Ace of Spades." Having a hand with two aces doesn't make it any more likely to have "an ace" than having a hand with one ace does.
This pretty much explains the phenomenon.