How, in the practical sense, does this work on other machines then? Am I supposed to carry my public key around with me everywhere, and load it on my friends mobile device just to check my email?
The dreamy eyed answer to this is that you would have your private keys on your phone and other devices would be able to ask it to verify a token. As seamlessly as possible and/or sensible.
How does a smart card help? So every device would need to be able to read a smart card? And third parties would need to be able to accept, store and present whatever is on the card?
Yes, every device would need to be able to read smart cards, and everyone would need to have smartcards. Other than the card, everything would just look like normal public key authentication; third parties would not need to store anything other than what they would store if this were done entirely in software.