Horizontal component of the electric field of an infinite uniformly charged plane

[A.T.]

On page 12, figure 9, you have the curves of the normal and radial components of the field as a function of altitude. You'll see that the radial component is not at all negligible (for example, at 0.5R from the center, the radial componet is 1/3 of the z component).

Indeed, I ran a simple test, and got similar numbers.

You can also look at figure 11 and 12 which shows that in the case of the parallel plate capacitor, the radial components of the two faces cancel each other out in pairs so that the field between the two disks is significantly more uniform than the field created by a single disk.

Yes, that makes sense.