Freewill

Balance is just one of the things that will make a tire shake. A tire that is out of round or one that has plies that don't run true with rotation will also cause shake. Balancing won't help with those problems.

A bent axle or spindle will also cause shake.

The steering stabilizer doesn't help with shake, and it isn't intended to.

Edit: Forgot to mention that running a tire out of balance for a while can wear the tire out-of-round. Pretty hard to do much about it except shave it or replace it. Because its a 4x4, all 4 tires should get shaved the same amount.