By the way, if anyone is still interested, at the time of this writing there
is a way to make even the latest version of Steam client run with Windows 7. You'll need to jump through some hoops to get it running, like installing
VxKex (whick will also let you run some other Win10-only programs on 7) and then disable some hardware acceleration options in Steam's Interface settings so that its UI doesn't freak out at you, but in the end it will work like a charm. You would still want to disable automatic updates after doing this though - there is no telling how long this method will work.
This guide worked for me (installing Legacy Update isn't necessary):