My proof feels weird to me because I on the way prove that an Artinian ring with trivial nilradical is a product of division rings, and I can't find this result anywhere, so even though I am quite confident in my proof, it is giving me doubts.
Because $Spec(R)=Spec(R_{red})$, we can assume $R$ has no nilpotent elements. Let $\{p_i\}_{i=1}^n$ be the set of all prime ideals. Because $Spec(R)$ is discrete every prime ideal is maximal, in particular two distinct prime ideals are coprime. Also the intersection of all prime ideals is 0. So by the chinese remainder theorem:
$$R\cong \prod_{i=1}^n R/p_i$$
Which is a product of division rings (because the $p_i$ are maximal), a finite product of division rings is clearly Artinian, finishing the proof.
Is this proof valid? Or is my hunch that I did something fishy, and if so how can I fix my proof? (note I am not assuming commutativity).