The GC as a straight line is only an approximation and only holds over arcs of about 20-30 degrees before it will noticeably deviate.
As an example; the equator is a great circle but will appear as an arc on a Lambert chart unless the standard parallels are at equal lattitudes N & S of the equator (in which case it is a cylindrical rather than conic projection).
In pure geometry the great circle is a plane that intersects the centre of the earth. If the area of interest is near the standard parallels of the Lambert, then it will be orthogonal to the cone in that region and hence appear close to a straight line. But if you look at the antipodean point of the great circle, it will intersect the cone at an oblique angle and the locus will be a distinct curve.
In simpler terms, it only holds true if you remain close to the standard parallels.