There would be no need for the chain to be perfectly clean. The paraffin will act as a solvent when hot. Any oil will dissolve in the paraffin and become trapped. It might even act to enhance the lubricity of the paraffin as well as make the paraffin more flexible. Any grit will fall to the bottom of the mixture and be trapped there.

Having said all that, that's the point of solvent carried wax based lubricants. The wax is dissolved in a solvent. After the solvent evaporates, the wax stays behind. Most wax based lubricants use soft waxes (petroleum jelly is an example) as well as harder waxes in the mix to make the wax more flexible and less likely to flake away as can be the case with waxes show in this thread.