I know they all connect into a single system, I mean is there one city or area that is directly served by all seven of them.
Looking at their route maps on all their wikipedia pages (and also using this), I actually don’t think there is. The closest you get is Chicago which has every railroad except KCS (the closest it gets is Springfield), and New Orleans which has everything except Canadian Pacific (which doesn’t come any further south than Kansas City).
So the answer to the question is no, they don’t.
But a drive around the Midwest would probably allow you to see all seven in one day. Hmm…