Would adding a fairing to landing gear to spin the tires before landing yield enough benefits?
Check out this answer to a similar question. What it boils down to is that pre spinning gear has been looked into but there are implications of spinning up that kind of mass. The rotational inertia created created by doing so effects the handling of the aircraft. On a similar note its often easier to have a simple part that wears out and needs to be replaced often (like a tire) than a complex part that needs maintenance and is more likely to be problematic.

The part you describe is pretty common on small aircraft to reduce drag over the wheels. As far as I know it does not aid in spinning the tires. Since the tires are more or less flush the air simply flows over them.
  

For what its worth these boots also can get in the way. If the plane lands hard the tires will balloon out and rub up agains the sides of the boot (depending on the tolerances) this may break the boot the tire or both.