I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith.pdf) is sufficiently straightforward except for Lemma 1 (located about halfway through page 3), which proves that the map $$\pi_{*}(MU_{(p)}[v_1^{-1}]) \rightarrow MU_{*}(K)_{(p)}$$ is faithfully flat.

However, it does so with incredibly powerful tools that would not be reasonable to include in expository work (i.e. the moduli stack of formal groups and all the crazy commutative diagrams that follow).

How can I prove that the required map is faithfully flat without using tools reminiscent of chromatic homotopy theory?

I'm choosing to follow the Mike Hopkins proof as opposed to Snaith's original formulation because the techniques are more straightforward and are motivated by more modern developments in Algebraic Topology.