It's probably a bad idea to authenticate on a public machine to begin with. How can you possibly know what that machine might do with your account once you log in?

But let's say you want to do it anyway. There is no need to memorize or type the entire key; a site-specific private/public keypair can be derived from a memorized master password and the domain name (for uniqueness) via a KDF. The site never needs to see the master password, so in the event of a breach all your other logins remain secure, and there are no key files to synchronize, just a single password for access to all your accounts.

You do need to trust that the PC you're using to log in is secure, though, since it has access to the master password; I'm not sure how you would get around that without involving some other trusted piece of hardware, and if you have that you could just use it to access the site instead.