This question is motivated by the following theorem: "If $X$ is a Riemann surface (connected, Hausdorff, holomorphic charts), and there exists $f: X \to \mathbb{C}$ non-constant, then $X$ is second-countable."
This statement feels strange/unsatisfying to me as it can only be applied to the non-compact case (every holomorphic function to $\mathbb{C}$ on compact complex manifolds are constant). Moreover, in this case, it is not obvious to me why there should exist non-trivial $f$. I see no reason why it should be true or why it should be false.
I was thinking of attacking this question using uniformization. Please correct me if this is incorrect, but according to uniformization, since $X$ is open, $X$ is either a quotient of the half-plane, or $\mathbb{C}$ itself. In the latter case we are done by taking the identity map. If $X$ is a quotient of the half-plane, then our goal is to find a non-constant holomorphic function that is preserved by the group action. I am not sure where to proceed from here, as in general, there are group actions where no non-constant holomorphic function can be preserved (compact quotients of the half-plane).
If anyone could provide a reference to an answer of this statement, some intuition, or an example of an open Riemann surface with only constant holomorphic functions that would be great!