1.) What you're proposing is against current USATF rules, which limit you to one uniform manufacturer's logo of a certain size and a club name of specific dimensions (and I'm all for changing this rule).
2.) How would these teams work? Are you talking about forming teams based on current shoe company sponsors (Team adidas vs. Team Nike vs. Team Rbk...) or about creating teams of higher value (Team Coors vs. Team Citibank...)? How does that fit in with qualifying for your national team and running at the Olympics/Worlds (where everyone must where a nationally issued uniform w/o any additional logos besides the kit manufactuerer)?
The big distinction between sports that do this well (NASCAR, for ex) and track, is that NASCAR's biggest event of the year is a private grand prix event (correct me if I'm wrong, I don't follow auto racing). Whereas track's biggest championship event is an international championship drawn on national lines, where teams are decided by citizenship.
Nationalism works well in most other countries (which is why the World Cup is huge everywhere outside of the US and track, while not at the top, is generally more popular than it is in America), but trying to create enthusiasm for a sporting event based on nationalism doesn't work anymore in the US, not since the end of the Cold War. What gets bigger ratings: the World Basketball Championships or the NBA Finals? Americans appreciate intranational, regional sporting rivalries (NY v. Boston, LA vs. SF, etc) far more than international matches.
How would you deal with that?