Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

One of the things that makes a cone simpler than a cube is that it is an “algebraic object” that can be defined by a simple polynomial identity ($x^2 + y^2 - z^2 = 0$). Taking cross-sections preserves this algebraic nature (since an infinite plane is also algebraic) so we end up with a quadratic curve in two variables, which is a fairly nice object. In some sense these are the “simplest” possible shapes beyond straight lines.

At the same time, you are probably aware that the shape of a quadratic curve varies drastically depending on where the minus signs live. The equation of a cone has enough minus signs that it is able to represent pretty much the entire spectrum of such curves, in contrast to, say, a sphere.

Lastly, there is some extent to which we study these because they were studied classically by the ancient Greek geometers. This kind of speaks to the utility angle in that there would be more applications of things that are well studied. But the preceding two points show that there are objective (non-historical) reasons to consider conic sections interesting. They are simple enough to be studied very thoroughly, and this simplicity also increases the chances that they would emerge naturally in many situations.

Elementary algebra begins with the study of linear equations. Quadratic equations naturally come next. Their graphs are ellipses, parabolas and hyperbolas. The Greek geometers knew them as sections of a cone. That geometry is often left out of beginning math courses today.

The quadratic functions are particularly useful in physics. I think they are not nearly as useful in other applications, or in understanding numbers in the news. When I teach mathematics to students who are not planning to go on in science we work on exponential functions instead.

My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some other 3D (or any dimensional) object?

Because there's nothing else. Look at the common 3D solids. The cross sections of a cuboid (and in fact any other polytope) are just a bunch of straight lines connected together, so this is just a piecewise linear graph, and piecewise linear graphs don't need to be motivated as cross-sections of three-dimensional objects. The cross sections of a spheroids (ellipsoids) are ellipses, which come up in cross sections of cones anyway. The cross sections of a cylinder are either trapeziums or ellipses. These are much less interesting and rich than conic sections!