It isn't necessarily any worse for public/cafe computers. Just like any login system, the UI should have the ability to remember you or not.

With the browser acting as your user agent, it can manage multiple contexts (work, personal) and let you choose which is sent to each site, without the need for separate browsers for each.

As an added tangent, consider that browsers are one of the few places where we can do things like "temporarily log me out of everything" (Incognito Mode / Private Browsing). They have some level of sophistication in session management already.