A common proof that an algebraically closed field has trivial Brauer group goes something like this:
Take $D$ a finite central division algebra over $K$, where $K$ is an algebraically closed field. Then for any $d\in D$ the field $K[d]$ is properly defined and makes for a finite extension of $K$. $K$ only allows for trivial extensions, so $D=K$.
If I'm right, (please tell me if I'm not), the fact that $D$ is central does not play any role in this proof. So what we basically proved is: there are no finite division algebras over $K$, so in particular there are no finite central division algebras.
(I've noticed something similar in a (different) proof where algebraically closed is replaced by $C_1$.)
I now have two questions.
First: Am I right in saying what I've said about the proof shown above?
Second: Are there examples of fields with a trivial Brauer group, which do have non-trivial finite (not central) division algebras over them?
Your proof is correct and yes, you don't need centrality here. See for instance Gille-Szamuely's book for a more precise treatment.
Essentially, you want to determine Brauer classes of central simple algebras over $k$ which split over a certain finite Galois extension $K/k$. But you know that each Brauer class contains an unique division algebra (up to isomorphism) over $k$. As you pointed out, there are no division algebras over $k$ except $k$ itself.
For an example of field with trivial Brauer group but still having a non-trivial division algebra over it, I'm not completely sure. Think about an algebraic extension $K$ of $\mathbf{Q}$ containing all the roots of unity; this has trivial Brauer group (a proof can be found in Serre's Local fields) but the field $\mathbf{C}$ is a division algebra over $K$ which is not central.