This is more or less a restatement of Philip’s answer. Hopefully, in a more accessible way for you.
First, the lines radiating outward from q aren't field lines per sayse. They are actually a graphical representation of E's vector field along the axis.
Second, the vector field drawn is for $E_q$ only. It's easy to see that the magnitude of vector field is dropping off with distance prior to passing through the plane. As you point out if the surface is a conductor, the magnitude of the E-field would be zero at the plane (and on the other side due to shielding the effect of conductors).