I found this statement about proving the existence of one-forms on a Riemann surface in an answer on MathOverflow:
This deep fact is essentially the same as the uniformization theorem. The problem is how to construct at least one holomorphic or meromorphic form with prescribed singularity. All known proofs use some Analysis, and none of them is simple. Once you have Uniformization, it is easy to construct holomorphic forms.
Why is this easy? Uniformization tells us that any Riemann surface has a universal cover that is $\mathbb D$, $\mathbb C$, or $\mathbb P^1$. It is easy to transfer forms from the base space to the covering space by pulling back. But how do we use the covering map to construct forms with prescribed singularities on the base space?