The typical solution of the typical polynomial ODE is uniformized by the Poincaré disc not by the complex line.

Indeed, after the work of McQuillan, it is known that the existence of a non-algebraic leaf uniformized by $\mathbb C$ imposes strong restrictions on the polynomial vector field. It turns out that there exits a projective surface birational to $\mathbb C^2$ where the foliation defined by the vector field has at worst canonical singularities and its cotangent sheaf has Kodaira dimension zero or one.

Anyway, we do not need the typical polynomial ODE to construct an example of a dense $\mathbb C$ in $\mathbb C^2$. It is sufficient to take $v = \lambda_1 x \frac{\partial}{\partial x} + \lambda_2 y \frac{\partial}{\partial y}$ with $\lambda_1/\lambda_2 \notin \mathbb R$. The typical orbit is dense in $\mathbb C^2$.

This is more a remark than an answer.

The typical solution of the typical polynomial ODE is uniformized by the Poincaré disc not by the complex line.

Indeed, after the work of McQuillan, it is known that the existence of a non-algebraic leaf uniformized by $\mathbb C$ imposes strong restrictions on the polynomial vector field. It turns out that there exits a projective surface birational to $\mathbb C^2$ where the foliation defined by the vector field has at worst canonical singularities and its cotangent sheaf has Kodaira dimension zero or one.