But my question remains the same: is there a direct way to prove the claim of the paper in the case when 𝑋 is the affine 𝑛-space (or even the affine line) over 𝑘.
Yes, read the proof of Proposition 1 in "The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture" by Alexei Belov-Kanel and Maxim Kontsevich.