Huygens principle and path integral for classical waves

I want to better understand what the path integral is and what it isn’t.
Even though I do this to learn QFT, this question is purely concerned with classical fields, no quantization is intended at all. Neither am I talking about classical particles with well-defined paths, though.

Feynman introduces the path integral with the Gedankenexperiment of expanding a double slit experiment to have an infinite amount of screens with an infinite amount of holes in each. He does this, talking about the options a particle has to travel to the screen. In QM, all possible paths then interfere and the path integral is used to calculate this interference.

What I read out there is nothing but Huygens principle.

Thus, if we have a classical field with a point source at point $A$ at time $t_0$ and we want to calculate the amplitude/phase of the field at point $B$ at time $t$, would the path integral be a proper tool for it?

If so,

• how does it relate to Fermat’s principle in this case?
• is there any important difference in either the measure or the involved action between a classical field or a quantum particle? (differences that would apply to any classical field)

Or put differently: Is the path integral a (mathematical) tool for quantum things or for wave things? Is it linked to the probability aspect or only to the wave nature of quantum phenomena?