Why would they? All of the probability-theoretic structure comes from the fact that a measure is not just an arbitrary one-form, and has lots of extra propertie

Why would they? All of the probability-theoretic structure comes from the fact that a measure is not just an arbitrary one-form, and has lots of extra properties. And then if you want to do probability over arbitrary manifolds, you need to talk about differential forms in general: there's no point at which it makes sense to generalize to one-forms but no further.