Earlier quoted context omitted.
> Secure and robust > value stored on-card These two things don't go together.
It could work as a sort of cryptocurrency. Cryptocurrencies are functionally equivalent to "stored value" and yet they remain secure. Plus, since we're talking about a government-issued currency, a lot of the drawbacks of a typical cryptocurrency wouldn't apply (the government can be considered the central source of trust, so no need for a proof of work system or anything).
Once you are in the domain of "Anonymous $1 cards which people swap with each other", you no longer have this option. The obvious thing is to have corresponding entry in online database, but then anyone who wants to verify the card value will have to access this database -- so your privacy disappears.
I am not saying that it's impossible -- people are smart, something can likely be figured out -- but it is not obvious to me those cards can be made harder to forge than cash.