If you really treated them as points the theory would blow up because the electrostatic potential energy of a point charge is infinite. This is dealt with by “renormalization” which is roughly: assume the theory isn’t really valid all the way to a zero length scale and that we can average out everything that happens below some cutoff size and that it doesn’t really matter where we place the cutoff because the theory works the same if you change the cutoff and change the other parameters accordingly.