I have a situation where a browser is constantly shared between different people. We don't want passwords remembered or logins to persist. However, this isn't a situation where it's appropriate to put the entire computer in lockdown (kiosk mode, Deep Freeze, Steady State, etc) — we have enough trust here, so that isn't necessary.

Browsers have incognito mode though. Can't you make incognito mode work ALL THE TIME? Yes you can: