Nope, I wouldn't say it's carried solely/primarily by the dielectric, since the material the conductor is made out of also matters when you are considering losses. (Someone correct me if I'm wrong.)
Also, you've got this weird thing called skin effect where the current mainly flows on the surface of your conductor. 8.5mm deep for 60Hz, but 2μm for 1Ghz. So what's in the center of your conductor doesn't really matter.
However, if you want your signal to travel at c, surround your conductor with vacuum instead of insulation. I think to actually reach exactly c your vacuum would have to cover an infinitely(?) large area around it.
If you want something more practical, air has a relative permittivity of 1.0006 (vacuum is 1.0), so if you surround your uninsulated conductor with air, you get a velocity of 0.9997c.
What if it's something in between, say cross-section is not O-shaped, like in a hollow wire, but C-shaped or almost closed circle? Where one side is vacuum and another a dielectric.
Thinking about it I believe it would be one of three possibilities: 1 slow the signal to the slowest side, 2 increase circuit resistance, or 3 "smear" the wave, so a short sharp signal would arrive long and dull (increased reactance?).
Yes, if the conductor is poor, it will have losses -- but still it is not the material by which power is conducted to the load. (Rather, because its resistivity creates an E field along its length and within it, you now have a nonzero Poynting vector within the conductor -- one which points outward into the environment!)
Not what's within each wire, but what is between the pair of wires is what matters. (Assuming of course the wires conduct well, as BenjiWiebe points out.)
And yes, using air instead of dielectric results in signal velocity near c. (A good example of this is ladder-line.)
