Better explained by the fact that tires are designed to fling the water out to the sides and therefore out of the “tracks.”
The tire does briefly heat the road under it, which is why you slide on ice. The tire melts the top layer of ice, and that water makes it easy to slide around on ice. The same mechanism is what causes you to slip while walking on ice. When it’s extremely cold, ice actually isn’t very slippery when you walk on it because the pressure of your weight can’t melt it before you’ve taken your next step.
The heating is more from compression due to the weight than from the tire being warm. Any time you change something’s shape, you end up with heat as a byproduct (which is why it’s always so funny in movies when they want to show someone is strong, so they have him bend a steel bar. Thing would be red hot if you bent it that fast). You’re pressing down on the road with a 3,000 pound plus vehicle - that’s gonna heat it up too. But unless you’re driving something like NASA’s crawler transporter, the heating isn’t going to be all that significant, especially since you’re probably moving over any given section of road at a pretty good clip.