> 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](https://arxiv.org/abs/math/0512171)" by Alexei Belov-Kanel and Maxim Kontsevich.

If you are only interested in the $n=1$ case $A=k \langle X,Y \mathrel| XY-YX=1 \rangle$, then give $X$ degree $1$ and $Y$ degree $-1$, so that $A$ becomes a $\mathbb{Z}$-graded ring. It is even strongly graded, meaning $A_1\cdot A_{-1} = A_0$, 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 there exist homogeneous central identities, a graded Azumaya algebra is a genuine Azumaya algebra. Done.