Why is that the extended real line $\mathbb{\overline R}$ do not enjoy widespread use as $\mathbb{R}$?

With respect to $\mathbb{R}$, the extension $\bar{\mathbb{R}}$ lacks properties like: being a group, a ring, and a field.

There are two meaningful extension of $\mathbb R$: one with a single $\infty$ point and one with two points: $+\infty$ and $-\infty$. Sometimes one is better than the other... so there is not a clear choice here.