A central part of the Rayleigh-Jeans law is that the number of allowed states follows $dN \sim \nu ^{2} d\nu$ because the allowed frequencies have to be standing waves. Then by equipartition we get that the energy density in a given frequency band increases without bound for increasing $\nu$, which is obviously unphysical.
Regarding a non-cavity solid object emitting thermally, by contrast, I'm picturing a lattice of atoms and electrons oscillating and/or colliding in such a complicated way as to be random. Perhaps there are conduction electrons, perhaps not. Now, I don't believe the phonon motion contributes much to EM radiation (?); instead, I think, electron oscillations (classically; in QM this would be transitions) dominate. Such electron motion would not be characterized by standing waves or frequencies which depending on the dimensions of the solid, surely?
It seems reasonable that at a given temperature, random "dipole" (or whatever) oscillations in a solid object would classically follow something like a Boltzmann distribution with a maximum at a finite frequency. Intuitively, this could lead to an emission with the qualitative shape of the experimental blackbody curve. Is there a classical theory of random thermal oscillations in a lattice which is relevant here?