I totally agree the Euclidean setting case is kind of trivial. I am asking now if there is some characterization on Riemannian manifolds so that one can be always able to find such a foliation (think about closed manifolds like the two- or three- sphere, they do not admit a foliation by convex sets either). (Thanks for your answer by the way)

You are right, I had in mind at least some strict convexity

I guess that density is the most difficult part to avoid with a counterexample, also because we are considering with respect to the (weak) notion of topology induced by the flat norm. May I ask you in which sense would it be helpful to restrict to the two delta currents?