Separating variables is a shameless method in mathematics or physics. There are no special rules in application. Generally speaking, if the physical phenomena described by differential equations have fluctuations and reflections, and their boundary conditions can form eigenvalue problems, then the method of separating variables can often work wonders. Personally, it depends on experience.

The general solution of the general equation can be obtained by the current superposition of the solutions obtained by the method of separating variables. As far as the differential equation I am in contact with at present is concerned, there is no missing solution in the solution obtained by separating variables.

Green's function is generally aimed at non-homogeneous differential equations, and it is often meaningless to separate variables.

Differential equations are basically insoluble, and only a few can be solved. Personally, I think they are all tried by predecessors at will, so it is ok to learn from others. I have been entangled in various solutions before, but now I am exposed to many differential equations and find that everything is a cloud. There are many more shameless methods than separating variables from Green's function, so enjoy it.