Why don't dielectric materials have coloured reflections like conductors?

The short answer is: Because metals are really absorptive (which comes from the fact that the nearly free electrons in the metal follow the oscillations of the radiation thereby depleting its energy), but some only in part of the visible range.

The reflectivity of a material is given by the Fresnel equations in terms of the index of refraction. They describe the angle dependency and further tell you that the higher the difference in the index of refraction the more light will be reflected at the interface.

It is important to understand, that the index of refraction in general can be a complex number. The imaginary part of the index of refraction describes the absorption of the material, while the (well known) real part describes the usual "optical density" causing refraction. So there are to possibilities for a material to reflect strongly: Either because it has a large real part of the index of refraction (like diamond) or because it absorbs light strongly (like metals). The latter effect can also be seen with lines written using a dark overhead transparency marker: They reflect in the colour range that does not pass through.

So, the reflection on the surface of metals is mainly due to the imaginary part of the index of refraction (that is, the absorptivity). For coloured metals like copper or gold the so called "plasma frequency" of the metal above which the metal begins looses its strong absorptivity is in the visible range (or in the near UV). Therefore such metals only reflect a portion of the spectrum well you get a tinted reflection.

The other materials (plastic, glass, apples) have one thing in common: They have a relatively low absorptivity (while for metals the wave only enters a few nanometers, the other materials range from transparent to waves entering at least several micrometers; the absorption caused by pigments in the material is typically much weaker than the one in metals). This means that the reflection is caused by the change of the real part of the index of refraction. As most materials are only slightly dispersive in the optical range, this means that all frequencies are reflected more or less equally, therefore the reflection is not tinted.

Plastics and biologic materials are very similar to the paint example you gave: they are basically transparent (well, translucent), and their color come from pigments inside. So the first air/material dioptre the light encounter is (mostly) color-neutral (at least for smooth surfaces), and specular is caused by the Fresnel term on this surface. Diffuse color really comes from the light entering the medium and diffusing once or more on pigments (+ligand, possibly).