From a technology standpoint, traditional paths are not designed to "share points" in the way you are looking for. A path is a series of points in a specific sequence...informing the logic of the path stroking code (and there are things like "Winding Fill vs Odd-Even Fill" if you want to dig into deeper technical implications.)
What you are seeking is more in the spirit of a mathematical concept of a Graph than it is a Path. You would like to think of your object as a collection of points, where each point doesn't lead lockstep to another...rather, it specifies a reusable coordinate that may be used for multiple connections without having to lead that path (or another) back to specify the point again.
I feel your pain; it's definitely how I think. If you haven't seen demos of things like SolidWorks then take a look. It shows how much better a program like Illustrator could be, but isn't. :-/
I've seen the SDK for Illustrator...and given the foundations they're working on, I wouldn't expect such features anytime soon. One feature that might make some aspect of your life easier, at some point, is the the "Live Paint" feature. Let's take two rectangles with no fill and overlap them a bit:
Select them both, and then go to Object -> Live Paint -> Make. After you've done that, pick the Live Paint selector tool (Shift-L):
Armed with this, you can start selecting the lines and shapes created by the intersections...deleting edges and fills that aren't part of your "specification". For instance, you could select the encroaching portion of the smaller square here with one click and delete it all at once...
It helps you manage points of intersection cleanly, but unfortunately doesn't create coordinates shared between paths. It's a thin layer over the existing format. Might still come in useful, sometimes.