When deriving the transformation law for a vector $V^\mu\partial_\mu$ under a coordinate change $x^\mu\to\tilde{x}^\mu$ for vectors in the tangent space to a manifold at a certain point, we arrive to the following rule:
$$\tilde{V}^\mu=\dfrac{\partial \tilde{x}^\mu}{\partial x^\nu}V^\nu$$
Imagine we have a vector field $V^\mu(x)\partial_\mu$. What does this vector field look like in the coordinates $\tilde{x}^\mu$? We can compute:
$$\tilde{V}^\mu(\tilde{x})=\dfrac{\partial \tilde{x}^\mu}{\partial x^\nu}V^\nu(x)$$
However, the RHS of this equation is a function of $x$, meanwhile the function on the LHS is a function of $\tilde{x}$. Thus we should write better
$$\tilde{\tilde{V}}^\mu(\tilde{x})=\tilde{V}^\mu(x)=\dfrac{\partial \tilde{x}^\mu}{\partial x^\nu}V^\nu(x)$$
Am I right in this reasoning? The problem arises because applying the formula $\tilde{V}^\mu(\tilde{x})=\dfrac{\partial \tilde{x}^\mu}{\partial x^\nu}V^\nu(x)$ for a vector field defined in polar coordinates $V^\theta\partial_\theta+V^\phi\partial_\phi$ does not yield a vector field $V^x\partial_x+V^y\partial_y$ where $V^x$ and $V^y$ are functions of $x$ and $y$. Instead, it yields
$$V^x(r,\theta)\partial_x+V^y(r,\theta)\partial_y$$
We can of course substitute $\theta$ and $r$ in this equation by using the inverse coordinate transformation, but this is not explicit in the formula $\tilde{V}^\mu(\tilde{x})=\dfrac{\partial \tilde{x}^\mu}{\partial x^\nu}V^\nu(x)$. Why is this the case? This formula tells us that we should directly get a function of the new coordinates.
This problem does not arise when dealing with a vector, since we just evaluate a certain point, but it does when dealing with vector fields. I feel this question is somewhat basic, but I am unconfortable and cannot get my mind around it.