I have some confusion regarding covariant vectors vs. scalars under coordinate transformations. It probably has to do with what it means to "undergo a coordinate transformation".
Under a coordinate transformation, a covariant vector has components that transform in the same way as the basis vectors. And a scalar is a quantity that is invariant under coordinate transformation. For example, see: https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors
Consider a specific point on a manifold, and an object $\partial_0 \equiv \frac{\partial}{\partial x^0}$. When transformed to the primed coordinates, $\partial_{0} \rightarrow \partial_{0'} = \frac{\partial x^\mu}{\partial x^{0'}}\frac{\partial}{\partial x^\mu} $. So it appears that $\partial_0$ transforms as a covariant vector. This seems to make sense, because $\partial_0$ is a part of the standard basis of the tangent space at that point, so it should transform as a basis vector.
However, the same object can be written as $\partial_0 = V^\mu \partial_\mu$, where $V^0=1$ and the other 3 components of $V^\mu$ are zero. This is evidently an invariant scalar, because we have raised and lowered indices that are matched. So $\partial_0$ should be a scalar and there seems to be a contradiction.
I think the issue is whether we want to insist that the object $\partial_0$ should remain invariant under coordinate transformation. If we do insist that it must be invariant, then $V^{\mu'}$ must adjust themselves to ensure that $V^{\mu'} \partial_{\mu'} = V^\mu \partial_\mu$. The alternative is that we think of $\partial_0$ as an element in the unprimed basis set, whose role is to span the tangent space. So then in the primed coordinate system there will be a primed basis set whose role is to span the tangent space, and the transformation is supposed to express $\partial_{0'}$ as a linear combination of the unprimed basis vectors. So in this case, the new transformed object $\partial_{0'}$ is not the same as the original $\partial_0$, and so $\partial_0$ transforms as a covariant vector.
Is this understanding correct, or is something else wrong?