@CharlesRezk If $X$ is an object of a cocomplete stable $\infty$-category $C$, then by "$X$ is small", I mean "$X$ is compact", which stably means that $Hom(X,-): C \to Spectra$ preserves coproducts (this is basically how the condition is usually formulated in a triangulated category). I'm going off of Theorem 8.5 of
Hovey-Strickland, but I suppose the gap in my reasoning is that I didn't actually check that $E^\ast(K(n))$ is finite.