Tyre wear and land gear shock reduced by wheel spin up
I don't believe there is problem with gyro precession in spinning wheels; it's more of a vibration issue.  On most airliners the brakes are applied automatically after liftoff to stop the mains and the main reason is unpleasant vibrations as the wheels slow down without the damping effect of being pushed into the pavement.  On less complex aircraft pilots are taught to tap the brakes after liftoff to stop the main wheels from shaking (to the extent that they do) as they wind down.   

Airliner wheels get statically balanced after tire installation but they don't usually get balanced dynamically like car tires, and there is often minor vibration that is not noticeable while rolling but can be quite noticeable while spinning on a gear leg dangling in the open when the wheel's natural frequency comes into phase with the gear, scaring the crap out of the pax. 

Nose wheels are typically stopped by a snubber in the nose gear well upon retraction so they spin down until then.  Being so much smaller, they aren't as big a problem but if the static balance is not good you can get a pretty noticeable vibration under your feet that grows and then declines as the wheels slow down, then stops as soon as the gear is up. If it's bad enough crews will snag it.

Anyway, to have all that mass spinning all the time the gear is out?  Even if there was a simple way to do it, without the usual weight/cost/complexity issues, airlines would probably be forced to dynamically balance their wheel assemblies and purchase hugely expensive equipement.

Cheaper to just let them wear down and recap them.  Aircraft tires can be retreaded indefinitely as long as the carcass is sound.