Dear David, I think there might be a slightly simpler proof of the analytic Quillen-Suslin theorem.

Given a holomorphic vector bundle $E$ on $\mathbb C^n$, it has a holomorphic connection since its Atiyah class $a(E)$ vanishes : indeed   $a(E) \in H^1 (\mathbb C^n, \Omega^1_X\otimes \mathcal {End}E) $ , and the whole cohomology group is zero by Theorem B. And then I remember that a long tile ago, Otto Forster explained in a talk an argument by Griffiths that you could deduce from that connection the holomorphic triviality of $E$. Extend each vector $v\in E[0]$, the fiber  of $E$ at zero , to to a global section $s_v \in \Gamma (\mathbb C^n, E)$ by first doing that on  the restriction $E|L$ of $E$ to lines $L\subset \mathbb C^n$ and then show holomorphic dependency on $v$ by invoking a theorem of holomorphic dependency of a system of differential equations on its paramaters (here $v$).This gives a trivialization of $E$. Unfortunately I don't have a reference for this method of proof  but I have no doubt you can fill in the details if you feel so inclined. However it is not clear how elementary this proof really is, since it invokes theorem B .

Finally, since you evoke the contrast for a manifold between  having its holomorphic line bundles trivial and all its holomorphic vector bundles of any rank trivial, let me quote :          
**Theorem (Forster-Rammspott)** On a Stein manifold of dimension $n$ every holomorphic vector bundle $E$ of rank $r\geq [n/2]$ can be decomposed as $E=F\oplus \mathcal O^{r-[n/2]}$ where $F$ is a holomorphic  vector bundle of rank $ [n/2]$ .

So we have the surprizing consequence that for a Stein surface the hypothesis that all holomorphic line bundles are trivial forces the conclusion that *all* holomorphic bundles, of any rank, are trivial.