I haven't tried to follow the math closely but should there not be some concern about the region of convergence? It looks like they don't specifically discuss it. Or is there some reason this isn't a problem in this context?

I fear they have completely overlooked it.

The Taylor series for the exponential is convergent everywhere so what radius of convergence are you talking about? All the functions they're approximating are convergent everywhere & you can easily prove that compositions of functions that are convergent everywhere are still convergent everywhere.