Brakes are an issue for almost any street car. Being heavy and fast makes it worse, but even something light and slow like a Miata will have brake issues on a track (not easy or cheap to solve either!)

The model S shares its cooling with the inverter the motor and the axle. I'm not sure if that is the limitation or the battery. The model 3 is different and there is video of one lapping ten or so laps at Laguna Seca without heat issues.

There is no inherent reason an electric car cannot be cooled enough to be a track car, the designer just has to make it a priority over say aerodynamic drag (and hence freeway range).

Tesla Model S/X don't have any cooling for the electric motor, because it's not needed for street driving. That's a significant issue for track driving, but not for drag racing.