What is assumed to be a more fundamental physical quantity in classical electrodynamics.
The charge density as a scalar field or the physical entity charge.
Well discrete charges, and in particular point charges, are a consequence of quantum mechanics. If you're considering just the classical theory there are no special conditions on the charge distribution. I'm not sure I'd say charge density was more fundamental than charge, but charge density would be what gives you the divergence of the electric field.
Physical charge is primary, though not free of problems. First I'll give the evidence in favor of charge as a scalar field.
Often the classical fields you are computing are the macroscopic fields. Just like when you talk about the density of a substance you don't intend it to jump to near infinity at each atom then down to near zero in between, you talk about an average over a large enough region that it smoothly changes from point to point (because percentage wise not many atoms net get in or out for a reasonable movement), but is small enough an average that you aren't averaging over different regions with different densities (you don't want to miss an impedance mismatch). So you do the same for electric fields and electric charges. They are called the macroscopic fields.
And that's almost always what you are being asked to compute in practice. And since you are basically defining these fields to have a smooth charge distribution, it makes sense to take that seriously physically. However, I've seen mathematical texts that have trouble saying exactly which kinds of charge density are allowed and disallowed, so it could be easier from a mathematical perspective to say that the fields are whatever nice enough things you want and then say that charge density is just proportional to the divergence (in whatever sense you want) of the electric field. I can see the seriousness of that approach too.
Those are the reasons for favoring the charge density as fundamental.
But there are problems. For instance you need a velocity field too if you want to have charges experience forces, and reasonable setup can develop catastrophes. For instance you can have some spherical shells of continuous charge with the inner ones pushed out harder than the farther ones, and get shell crossing, so the whole idea of a single velocity field coupled with a single charge density field breaks down. So you must consider charges with actual locations and velocities to deal with the shell crossing.
But even the velocity field itself doesn't make things as simple as you'd like. To get a current you need to take the velocity vector field and the charge density scalar field and multiply them, but if positive ions are going a different direction than negative ions, you need multiple charge density fields and multiple velocity fields. If they are accelerating at different rates, you need one for each charge to mass ratio, so basically a scalar field and a vector field one each for each species and isotope. You've almost gone over to the charge side already.
OK, so why not go over to charges. It's annoying, it's complicated (and more complicated than needed often), and there is a stench of dishonesty too. You don't want to imagine little fixed individual particles with prefect fixed positions and momentums, because you don't really think that is accurate (quantum mechanics) you wanted to talk about macroscopic things and you wanted it to be easier. You probably choose classical electromagnetism because you thought those individual small scale motions weren't important.
So charge is fundamental. But ignore it when you can get away with it. Use it when you have to, and then don't read too much into when you do have physical charges.
As a technical note, you do often build up a continuous distributions as if it were made up of mathematical point charges or mathematical perfect dipoles, you don't really think there are real dipoles with charges much larger than $e$ and with separations smaller than a nucleus. But if it simplifies the model enough as is accurate enough then go for it. But those are not the physical charges. But the math you use for the physical charges can be used to model those mathematical point charges and point perfect mathematical dipoles. And that's fine too if you don't read too much into it.