Modern notation has contributed greatly to making Maxwell's equations more understandable. I dare you to have a look at Maxwell's original paper (https://royalsocietypublishing.org/doi/10.1098/rstl.1865.000...) and try to understand it, I'd say even for a seasoned physicist it's not trivial. Compare that to the modern differential form of the equations (e.g. https://en.wikipedia.org/wiki/Maxwell%27s_equations), which (IMHO) are really easy to understand even for second-year Physics students (given that you understand the underlying vector operators).
Personally, I found the "flow analogy" always the most intuitive, and there are some books that teach electrodynamics in that way. Typically one would start with electrostatic problems and work one's way to the more complicated stuff like magnetic fields.
I guess to derive a "complete" understanding of classical electrodynamics you need to understand the concept of relativity as well. Magnetism is a consequence of relativity and the finite speed of light, so if you accept that it becomes easier to understand (IMHO). Of course you then have to "understand" relativity, which just moves the problem to a different area. But then again, it's always like that in Physics :)
Magnetism affects moving charges, but not stationary charges. A magnetic field in your reference frame is really just an electric field in the moving charge's reference frame.
Personally, I found the "flow analogy" always the most intuitive, and there are some books that teach electrodynamics in that way. Typically one would start with electrostatic problems and work one's way to the more complicated stuff like magnetic fields.
I guess to derive a "complete" understanding of classical electrodynamics you need to understand the concept of relativity as well. Magnetism is a consequence of relativity and the finite speed of light, so if you accept that it becomes easier to understand (IMHO). Of course you then have to "understand" relativity, which just moves the problem to a different area. But then again, it's always like that in Physics :)