Triviality of holomorphic vector bundles over $\mathbb{C}$

Question

Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle. I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and Holomorphic Mappings), which holds for contractible Stein manifolds and the other one can be found in O. Forster, Lectures on Riemann Surfaces, and applies to non-compact Riemann Surfaces.

Both these arguments are highly nontrivial, and I thought that in the special case of $\mathbb{C}$ there might be a simpler proof. For example, trying to specialize the second proof above, one finds that the case of $\mathbb{C}$ is implied by proving one of the following:

1. $H^0(\mathbb{C},\mathcal{M}(E))\neq 0$, i.e. $E$ has a non-zero meromorphic section;

2. $\operatorname{dim} H^1(\mathbb{C},\mathcal{O}(E))<\infty$.

Although I find them likely to be true, I was not able to prove any of the two. So, the question is:

Is there an easy proof of either 1. or 2. above? More generally, is there an easier proof than the two arguments above in the case of $\mathbb{C}$?