No, it doesn't need pulley. It's the left hand box on the bottom of the screen. However, more pressing issue would be that you would likely have to reload the page on previous tab when coming back. it should get you back to right place on the page though.

Don't know about pocket client.

Don't use twitter by myself... but from that description, it might be a good idea to tweak the client so that some functions are always available (like they are in browser). Not everything has to be in the pulley.

Another option would be to invent another gesture to access pulley from every position. Perhaps something like tap and pull? Such rationale is already in use on some apps, like gallery.