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
If you are only interested in the n=1 case $A=k \langle X,Y~|~XY-YX=1 \rangle$, then give $X$ degree 1 and $Y$ degree -1, so that $A$ becomes a $\mathbb{Z}$-graded ring. It is even strongly graded, meaning $A_1.A_{-1} = A_0$, so all graded info is determined by the zero-part $A_0=k[XY]$. In particular, if you divide out a graded maximal ideal you get a strongly graded ring with part of degree 0 a field and finite over its center, and these must be central simple algebras (in this case, just $p \times p$ matrices) proving that $A$ is what is called a graded Azumaya algebra. Now, for the fun part, as they exist homogeneous central identities, a graded Azumaya algebra is a genuine Azumaya algebra. Done.