PuTTY SSH auto login
Your approach seems good, however, there's one default setting in the behaviour that's causing you trouble here and probably making you believe nothing's happening.
However, when I try to do so a new window for PuTTY appears, but closes and exits instantly after login
4.1.3 `Close Window on Exit' Finally in the Session panel, there is an option labelled `Close Window on Exit'. This controls whether the PuTTY terminal window disappears as soon as the session inside it terminates.