> 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.

https://arxiv.org/pdf/math/0512171.pdf