Thanks for the comments, @OtisChodosh and Willie Wong! But even using an orthonormal frame I don't see how this follows from AM-GM. In an orthonormal frame we have $\|\text{Rm}\| = \sqrt{\sum_{i,j,k,\ell} (R_{ijk\ell})^2} \leq \sum_{i,j,k,\ell} |R_{ijk\ell}|$, but how do we combine $R \leq \sum_{i,j,k,\ell} |R_{ijk\ell}|$ with this in order to obtain $R \leq C_n \|\text{Rm}\|$?

@DeaneYang yes, but I'd be interested in knowing the incomplete examples too.

@IgorBelegradek By Myers's theorem, a complete Einstein manifold with positive scalar curvature is compact.

@WillieWong it should be $-kR$, to cancel out the $kR$ that appears earlier.