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, where $MU$ is the Thom spectrum for complex vector bundles, $\pi_{*}MU$ is the associated Lazard ring, and $v_1$ denotes coefficient of $x^p$ in the $p$ series of the formal group law associated to $MU$.
This map, combined with a previously established isomorphism from the Landweber exactness (also discussed in the paper), is used to show that $\pi_*R[\beta^-1]_{(p)}$ is torsion-free and concentrated in even degrees; essentially we're constructing a spectrum and showing that it is the spectrum representing complex $K$-theory.
However this map is proved to be faithfully flat 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.
If there are other proofs using more classical methods in Algebraic Topology (by this I mean not $\infty$-categories) that can be used to prove this, I would appreciate a pointer in the right direction (I looked at Snaith's original papers and they're not understandable to me)
