Earlier quoted context omitted.
But should all sites really be optimized for the user at a public library computer? At the expense of convenience for the large majority of users that are on a personal or work computer? Doesn’t make much sense to me. Also the computer itself solves this problem for you in many cases, a guest profile typically deletes all browser session info when you log out.
All sites? Probably not. Many sites? Probably. You're assuming people log out reliably or otherwise behave in the most secure way. They don't. I also don't see how logging out/killing a session after 15 minutes of inactivity is much of a hardship for the user.
And it's not just extremely annoying, it's also completely unnecessary. Just put a "trust this browser" checkbox on the sign-in page and adjust the session timeout accordingly.