Why does an electric field always leave a conductor perpendicular to the surface?

I know that the vector sum of all the electric fields at that point will cancel out so you are left with the one that is perpendicular but i dont think that's the answer they're looking for. I have a sort of an idea that it has to do with electrostatic equilibrium where charges distribute themselves so there is as little repulsion force acting on them as possible and if you have a parallel component then it implies that charges must accelerate in that direction which they don't so there can only be the vertical component. Is that vaguely close? Thanks in advance