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.
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.
That works. It defaults to the "safe" behavior, but allows users to self-select into other behavior that they find less objectionable.
FWIW, my end-users are using public computer labs, so we have to build for the worst-case in terms of user security habits.