A rubber piece over the axle would add some grip and increase its diameter, what would make it run faster along the track. But it shouldn't be needed to make it work.
Experiments with several crude but successful LEGO knock-offs of the Maxwell/Ptolemy top confirm some of the observations noted above...
1. Speed along the track increases with increasing stem diameter, as you said.
2. Speed along the track also increases with increasing stem-track friction, as that reduces slip at high spin rates. Still, metal-metal friction might be enough to get some of the effect.
3. Speed along the track also increases weakly with decreasing CM-tip distance, whether the CM is above or below the tip. So fastest when the CM is at the tip -- in which case, no precession bucking opposing stem motions along the track. (Hence the bell-shaped rotors.)
4. Nothing magic about a spiral track. S-shaped, circular, linear, and polygonal tracks also work well. Fingers, too.