I think the author uses minimal set because it allows constructing any sigma-algebra from it. This (minimal set-based construction) is only possible for finite spaces. It allows, for example, an intuitive explanation of how sigma algebra F2 is coarser than F1. It serves a similar purpose as Borel sigma algebras in the general case, although a minimal set does not need to be a sigma algebra.

Using different sigma algebras allows describing easily all events on which the probability is defined, and how those events may change with time (filtration). I am not sure how use of sigma algebras (or algebras for finite case) can be avoided in general.