One could always argue that since there is only a finite (even if unimaginably huge) amount of possible branching points, we're actually talking about a discrete probability distribution.

How do you mean?

I'm talking about the fundamental physics of the universe. From a mathematical perspective, it's far more elegant (ergo, more likely) to deal with a partial differential equation defined on a continuous configuration space. Attempts to discretize the space in the name of infinite-set atheism seem ad-hoc to me.