You don't need a metric to define the differential of a function,
and the cotangent bundle carries a canonical one-form.

But you do need a metric to define the gradient, and the
tangent bundle does not have a canonical vector field.

These are not difficult truths, but still... why the preference
toward "co"?

If you want to differentiate functions from a manifold to (say) the real line R, then you want to use the cotangent bundle on the manifold.

If instead you want to to differentiate functions to the manifold from the real line (i.e. parameterised curves), then you want to use the tangent bundle on the manifold.

So the preference comes from whether you want to use the manifold as the domain or as the range of the functions one is differentiating.
