Steam without any windows normally continues running with just a tray icon; this happens with the desktop client also (and on Windows/OSX). You can set STEAM_FRAME_FORCE_CLOSE=1 to have Steam exit more often on last-window-close.
Closing as this appears to be by-design behavior (a deliberate design compromise which is not perfect, to be sure).
Nothing extracted yet.
Steps to reproduce:
Desired effect:
Actual effect:
System:
The window manager is non-reparenting, which might be part of the problem.