1. If a linear space L and its dual L* are mutually dual, why should one care which object --vector or covector-- is used to define a particular physical concept?

If one has a linear space only then indeed there is a full symmetry between these two spaces. The difference is magically born at the moment one introduces differentiable manifold and the spaces L and L* "become" the tangent and cotangent space, respectively. One of the crucial differences is that differential forms have well-defined exterior derivative "d", while vector-fields have well-defined Lie bracket "[ , ]".

2. Since any manifold M can be embedded into some linear space Rⁿ, why bother with the definition independent of such an embedding?

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