When/how can you integrate tensors on manifolds and what does it mean?
I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all the tensors along the path to a common point on the paths and "sum" them there, but what about area, volume and higher dimensional integrals? What is required for that to make sense? Parallel transport does not seem sufficient anymore because there are no longer unique path to a common point and parallel transport is path dependent.
I can't really make sense of your idea for line integrals. Using parallel transport you can more or less reduce this to the case of integrating a tensor on an interval-but how do you do that? Normally integration should output a number, and for this it really seems you need a 1-form. Given a metric, it's true that you can do better: you can integrate functions globally using the volume form, and you can type-change to integrate $(k,\ell)$ forms where $k+\ell$ is the dimension of the manifold. But I don't see how you can do better than that.