Basically, you want to have a handly to speak of the limiting points "at infinity" and not add some disconnected copies or foliations of Minkowski space. In other words, the extra space you are adding by going from $R<\pi$ to $R\leq \pi$ can be properly thought of as limiting points of the original space, while going to $R<\infty$ adds a lot of extra space that cannot easily be mapped back to the original coordinates.
In terms of paths, an outgoing light ray in Minkowski space would asymptotically approach $R=\pi$, but will not reach any value $R>\pi$.