Classical *global *solutions are irrelevant to quantum Yang-Mills, you are confusing "classical solutions" (meaning a point in the classical field theory phase space) with "local classical solutions" (action minimizing local bumps, like instantons).

You can partially describe the random configuration of the Euclidean gauge field by sprinkling classical local solutions at random, and patching these local solutions together into a global approximate action minimum which serves as a basis for a near-minimum quadratic expansion. The key word is "sprinkled at random". The instanton gas is such a picture, so is Polyakov 2+1 monopole confinement, and the Argyres version in compactified 4d gauge theory, or any other soup-of-classical-solutions picture. The soup is what makes it a description of a gauge vacuum.

An instanton gas configuration is not a classical solution, a single instanton is. A configuration of an instanton gas is only an approximate solution, the approximation getting better when the centers are far apart, and it is made by superposing instantons, which don't superpose exactly (this isn't a linear theory).

These approximate classical configurations are not in any way related to precise classical points in field theory phase space, these are patched together local solutions used to approximate the vacuum by summing random configurations which are locally action minimizing. These methods can perhaps work to describe the gauge field at long distances, but they have no relation to phase space methods in 0+1 d quantum systems.

I am skeptical of classical methods, because the lattice picture is much easier and much more numerically accessible strong coupling lattice gauge theory, so I don't care about this. There's not much point to my mind in even considering the classical equations for quantum Yang-Mills, every useful classical-solution concept has a much easier analog on the lattice, but people like it, and it never hurts to learn about special solutions. They just like it too much in math, because they have all sorts of fancy specialist theorems about classical solutions, and not so many theorems about lattice configuraitons.