It doesn't imply that. The singularity is just the inflection point.

Singularity and inflection point are incompatible mathematically and in the plain sense, it really is focused on a particular moment and always has been, hence the term.

And it's definitely supposed to imply some kind of historical discontinuity not a change in convexity.

Which assumes the presence of an inflection point that keeps inflecting rather than revert to an S-curve. The growth model is not borne out yet to declare what shape it is.