What I'm curious now is how one could use software (even PowerPoint) to make graphs that replicate that handmade aesthetic.

matplotlib has an xkcd style for a different sort of handmade aesthetic. And as reluctant as I am to bring this up as a comment to this post, “plot xyz and make the graph look like it was published in the financial times (but without ripping off their visual brand)” is a remarkably effective prompt after a little tweaking and I imagine something similar would work for other styles.