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 $R>\pi$.

Basically, you want to have a handle 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 $R>\pi$.

Update: You can intuitively think of the limiting points of Minkowski space as those points where $t$ or $r$ "become infinite". (Note that the angular coordinates do not play much of a rôle here, so there could be basically any number of those.) Given that the original metric is $\text{d} s^2=-\text{d} t^2+\text{d} r^2+\dots$, these boundary points can be infinitely far away in space, $\int\text{d} s^2\to\infty$, or time, $\int\text{d} s^2\to-\infty$, or infinitely far out in affine distance along null geodesics. The conformal diagram map the infinite ranges of $t$ and $r$ to finite open intervals (e.g. of $T$ and $R$), and by adding the endpoints we have a well-defined way to talk about the "points at infinity" and how they relate to the causal structure of spacetime.