A proof can be extracted from Cartan's ["On manifolds with projective connections."][1] (Translated by D. H. Delphenich).
In section VII Cartan defines a projective connection in the projectivised tangent bundle ("manifold of elements") with connection one-form:
$$
\omega = \begin{pmatrix}
\omega_0^0&\omega^1&\omega^2\\
\omega_1^0&\omega_1^1&\omega_1^2\\
\omega_2^0&\omega_2^1&\omega_2^2\\
\end{pmatrix}
$$
After normalising the connection he arrives at the curvature two-form
$$
\Omega = \begin{pmatrix}
\gamma&0&0\\
\Omega_1^0&\gamma&0\\
\Omega_2^0&\Omega_2^1&\gamma\\
\end{pmatrix},
$$
where
$$
\begin{aligned}
\Omega_2^1 &= a\omega^2\wedge\omega_1^2,\qquad a=-\frac{1}{6}\frac{\partial^4f}{\partial y'^4}\\
\Omega_1^0&=b\omega^1\wedge\omega^2,\qquad b=\text{complicated function}\\
\Omega_2^0&=\text{complicated two-form}\\
\gamma&=\Omega_0^0=\Omega_1^1=\Omega_2^2=\text{complicated two-form}.\\
\end{aligned}
$$
In section 23 Cartan then states the dual connection with connection one-form
$$
\varpi = \begin{pmatrix}
\varpi_0^0&\varpi^1&\varpi^2\\
\varpi_1^0&\varpi_1^1&\varpi_1^2\\
\varpi_2^0&\varpi_2^1&\varpi_2^2\\
\end{pmatrix}=\begin{pmatrix}
\omega_2^2&\omega_2^1&\omega^2\\
\omega_2^1&\omega_1^1&\omega^1\\
\omega_2^0&\omega_1^0&\omega_0^0\\
\end{pmatrix}
$$
which is simply the primal connection transposed with respect to the antidiagonal and the same goes for the the curvature two-form:
$$
\Pi = \begin{pmatrix}
\gamma&0&0\\
\Pi_1^0&\gamma&0\\
\Pi_2^0&\Pi_2^1&\gamma\\
\end{pmatrix} = \begin{pmatrix}
\gamma&0&0\\
\Omega_2^1&\gamma&0\\
\Omega_2^0&\Omega_0^1&\gamma\\
\end{pmatrix},
$$
so that we have
$$
\begin{aligned}
\Pi_1^0=\Omega_2^1\\
\Pi_2^1=\Omega_1^0
\end{aligned}
$$
We have that $\Omega_2^1=0$ is the condition that the right-hand side of the differential equation is a polynomial in $y'$ with degree at most 3 and the condition $\Omega_1^0=0$ is then "infinitesimal desargueness". In the dual connection these two conditions are swapped.

It might be argued that "infinitesimal nondesargueness" should really mean that any of $\Omega_2^1$ or $\Omega_1^0$ is non-vanishing since they are both obstructions to the construction of homogenous coordinates which then gives a projective structure on the $x,y$-plane with the solution curves as lines. The availability of homogenous coordinates is equivalent to the Desarguesian property of a projective plane.

Here's a couple of blog posts discussing these things:

 - [The geometry of y''+y=0][2]
 - [The geometry of a second order differential equation, part 1][3]
 - [The geometry of a second order differential equation, part 2][4]

  [1]: http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/cartan_-_projective_connections.pdf
  [2]: https://curvedmaths.wordpress.com/2020/07/02/the-geometry-of-yy0/
  [3]: https://curvedmaths.wordpress.com/2020/07/03/the-geometry-of-a-second-order-differential-equation-part-1/
  [4]: https://curvedmaths.wordpress.com/2020/07/04/the-geometry-of-a-second-order-differential-equation-part-2/