The heart of the proof is good, and I wanted to comment that you could easily prove a "cousin" of that theorem for noncommutative Artinian rings!
I am, of course, using the noncommutative definition of prime ideals which generalizes the commutative one.
Proposition: A prime ideal in a right Artinian ring $R$ is maximal iff it is prime.
Proof ($\Rightarrow$) If $M$ is a maximal ideal, $R/M$ is a simple ring, which is certainly a prime ring. It follows that $M$ is a prime ideal.
Proof ($\Leftarrow$) (This is where the theme of your proof can be applied again!) Suppose $P$ is a prime ideal. Then $R/P$ is a prime ring. Since $R$ is right Artinian, so is $R/P$. But the Artin-Wedderburn theorem says that such a ring is simple, hence $P$ is maximal. $\Box$
The connection is that Wedderburn's little theorem is like the Artin-Wedderburn theorem: one says that a finite domain is a field, one says that a right Artinian prime ring is a simple ring. ("Right Artinian" is a weaker form of "finite" and "prime" is a weaker form of "domain".)