First, state income tax is exactly that, income tax imposed by the state, and governed by the state. There may be local jurisdictions that impose additional income tax but any savings there would be trivial.
While sports are important to the local economy, my understanding is that they are relatively minor at the state level.
Second, LeBron (and other athletes) earn playing income in every jurisdiction they play in, not based solely on the jurisdiction that the team is based in.
This means they are responsible for filing taxes in each of these jurisdiction.
In addition to playing income, they make money through endorsements and other investments. These are considered income in whatever state they claim is their residence.
It's in a players interest to establish a residence in an area with favorable tax laws.
Cities may try to woo elite players to join them, but tax savings isn't very compelling.