What is the benefit of using symmetric kernel in Fourier transform?
A set of symmetric kernels along Cartesian axes should make the transform consitent with respect to (at least 90 degree) rotations of the input.  If the 2D input yields a particular 2D output, rotating that same 2D input by an angle of 90 degrees, should yield the same 2D output rotated by 90 degrees.  If the kernels in the two dimensions were not symmetric, the transform would yield different output that could not be rotated back to match the original output.

I'd have to think if the invariance applied to angles that were not multiples of 90 degrees.

You may also want to have a look at the Abel transform which has a circularly symmetric kernel, useful for physical situations with circular symmetry.