... the electric field serves as a medium for the magnetic field to propagate and the magnetic field serves as a medium for the electric field to propagate in an electromagnetic wave
That what you are describing can be observed when an electromagnetic wave is emitted from an antenna. Electrons of an electrically conductive rod are periodically accelerated back and forth. During the accelerations a common magnetic field is formed around the rod. And then the electrons no longer manage to collect this field. Instead, an electric field is formed again from the magnetic field. And so on. The EM wave moves away.
This answers your question. You are right.
But, to understand why this happens, let's look at the details. The key point is how the energy - coming from the wave generator that moves the electrons back and forth - dissolves.
First of all, the skin effect is important. It turns out that the electrons that are accelerated in an alternating electric field are first and foremost those on the surface of the conductor. They are simply more mobile than the electrons inside the metal.
The next point is that the wave generator keeps pushing energy into the conducting rod. Of course some of the energy is transferred to the metal lattice of the rod. The rod gets warmer. But mainly, the common magnetic field of the electrons should go back into the electrons, or what will it do?
You have answered the question correctly. Around the rod there is another volume which is free of any matter. Here the magnetic field changes into an electric field again - this time away from the rod -, and escapes into the free from matter vacuum.
A note. The escape of electromagnetic radiation is constantly happening around us. Every body around receives and emits energy and comes into thermal equilibrium with the environment. Looking at the smallest detail, electrons absorb photons and during relaxation they emit photons again. The hotter body has more excited electrons and emits EM radiation of higher intensity, and the cooler body receives more of these photons than it emits.
The special thing about radio waves is the technical solution to accelerate many electrons at once. The forced oscillating common electric field allows primarily polarised photons with their aligned electric and magnetic field components to escape into vacuum.
This is a wonderful possibility to use or even measure these fields. About the progressive transformation of the electric and magnetic field components in the photon we can conclude from their common EM field in a radio wave! Hertz did it and found out what you also concluded in your question.
