If there were an ECDSA card out there, it sounds like a private key could not be extracted from it, even for backup purposes. Is this correct?
This depends on the implementation. It could be anything from the card generating the key and never revealing it, to the client generating the key and just storing it on the card in an insecure fashion. It's up to you.
It would even be possible to store the key weakly such that it requires a pin, and that brute-forcing the pin would take a week or so. That way it would provide some protection against loss or theft while you could always break in if you forgot the pin.
If the implementation were done on a non-ECDSA javacard, something like a computer or smartphone would still have to interface with it, correct?
A smart card would only be useful as a secure wallet, not a complete implementation.
I think multiple ideas here are viable and similarly good. I'll think about which one I'd want to implement and in the mean time, this discussion may strike up the interest of anyone with skills more applicable to either of these devices