Not sure what the best solution is, but some thoughts. First, you definitely want a different keypair per device.
One approach is to just supplement passwords. You could use a password (2FA, etc) to log in, then the site gives you the option of adding that device's public key and from then on you can log in on that device automatically. The site would maintain a list of public keys associated with your account, just like github does for repositories.
Of course, if you don't trust those 20 work computers, you wouldn't want it set up so that anyone using them can log in to all your accounts. One thing the browser could do is password-protect your private key, so you have to enter the master password when you start the browser, and as long as you remember to exit out of the browser, the next person to use it won't be able to use your logins.
