So why do we not see problems with EM waves permeating without a medium?Ah. That's an old discussion which we've had many times before! I'd hesitate to get too side-tracked into it now as there are so many other threads in which we've all growled at each other about it! It goes back to the whole question of why any concepts, like "wave" or "particle" or "energy" or "mass" are used in the first place, and the extent to which they are useful metaphors for explaining observations. etc.
Try constructing a picture of light travelling as waves in 3 dimensional space from several sources and you will soon see the problem.If you mean a problem of visualization I totally agree. It's impossible to visualize. One of the things that I find endlessly fascinating is the fact that wherever I am there is a constant silent cacophony of radio waves, flying around me in every direction. Every single radio and TV channel. Emissions from stars, planets and distant galaxies. Possibly signals from alien civilizations. All there, all the time, all at once.
Simple logic tells me that they must be there all of the time because of the principle that I could choose to intercept any of them whenever I want. All I need to do is to encourage them to cause a resonant vibration of the electrons in a piece of wire. All I need is a capacitor, an inductor and a resistor - the electronic equivalent of a damped pendulum - tuned (by picking an appropriate capacitance) to resonate with the frequency I want to detect.
And, even for a single signal, the continual pendulum-like to-ing and fro-ing as a varying electric field gives rise to a varying magnetic field and the magnetic field then gives rise to another electric field is fundamentally impossible to visualize in the way that it actually occurs. Of course, we can draw simplistic schematic one-dimensional diagrams of this. But they don't convey the fact that it is happening in 3 dimensions at every point.
Yes, visualizing the unvisualizable is always going to be difficult. All we can do is create many different imperfect reflections, in the form of diagrams, equations and descriptions. A bit like trying to gain an appreciation for the appearance of an object by examining many different images of its shadow.
Back to the subject, a thought..Energy is what matter does.Sounds promising. But if we visualize energy as an activity carried out by matter, how can we convey the observation that it appears to be possible to transfer energy from one piece of matter to another? (As snooker players often seem to do.)