Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the stable homotopy category) consisting of the $p$-acyclic spectra is generated (as a localizing subcategory) by the finite $p$-acyclic spectra? Is there a reference for this?

To be clear, I'm talking about Bousfield localization with respect to $\pi_*(-) \otimes {\mathbb Z}_{(p)}$. It is definitely a smashing localization.

Recall that a spectrum is $p$-acyclic if $\pi_*(X) \otimes {\mathbb Z}_{(p)} = 0$. Thanks for your attention.