# Why do we need conformal compactification to define the global conformal group?

For starters, several conformal transformations, e.g. the special conformal transformations, take finite points $p\in M$ to $\infty\notin M$, which technically violates OP's suggested definition.

For Euclidean space $\mathbb{R}^n$, the conformal compactification $\overline{\mathbb{R}^n}\cong \mathbb{S}^n$ is the one-point compactification, i.e. the $n$-sphere, which indicates that $\infty$ should be treated on equal footing with other points.

The notion of infinity becomes more subtle in the case of indefinite metric, and the conformal compactification helps resolve this, cf. e.g. this Phys.SE post.