Why don't evanescent waves give rise to electromagnetic waves?

An evanescent wave is a superposition of the incident (penetrated) wave and the medium reaction to this incident wave (a radiated everywhere wave). The interference is "destructive" in the medium direction and "constructive in the opposite direction.

You must consider all sources of a wave to understand its behaviour.

Indeed, an evanescent wave only has a wavevector real part parallel to the interface.

A propagating wave obeys $omega^2 = k_x^2+k_y^2+k_z^2$ with $k_i^2>=0$. For an evanescent wave at least one wave vector component has $k_i^2<0$. In this direction the wave amplitude decays exponentially, while it propagates in the other direction(s). Such solutions of the wave equation are only possible near an interface. As an example, total internal reflection at a glass-vacuum or glass-air interface is accompanied by an evanescent wave outside the glass.

The existence of such waves are a consequence of the Fresnel equations, which describe what happens at a plane interface between two different effective media, solely characterised by their (complex) indices of refraction. The phenomenon is unrelated to interference.