In general, the position vector indicating a point in space is given by:

$\vec{x}=x^{\alpha} \hat{e}_{\alpha}$

But if the basis vectors also depend on position, such as a curved space $\hat{e}_{\alpha}(\vec{x})$ isn't this an implicit equation for the position vector?

Defining vectors as something on the tangent space to a point makes sense, but position vectors indicating that point in the first place seems paradoxical.