This video claims to be able to generate a key pair directly on the Smartcard so the private key never leaves the card. Is this operation common to all smartcards, or do only select smartcards support this?
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A smartcard is just a chip on a card, so you probably want to ask about cryptographic smart cards.
I have never worked with cryptographic smart cards but that sounds like a completely reasonable functionality of such smartcards since they are intended usually for authentication and encryption which usally use private keys for that.
The advantage of cryptographic smart cards usually lies in that, that the private key is stored in it. That is intended to prevent the compromise of the keys because even if the computer, that reads the smartcard, is compromised the smartcard is safe and the private key is safe too.