"What does a team name get you?", it gets you criticized and bashed apparently, lol. Don't hate the player, hate the game.
A few very basic things it "does" get you.
Entry in "some" tournaments and events that not just any organization can get into. Obviously there are dozens of events that will let anyone with the fees in also.
That same organizations events can draw better teams and competition than some other organizations. So again, better competition for your player(s) when the org hosts its own events.
Go to a tryout or team practice and see what talent it draws vs some other organizations.
Go out and try to start a team with no name or lesser known organization and then compare doing the same thing with a well know organization and see the difference. You get better playing with other like players and talent. Not to mention experienced coaches, that can be in the eye of the beholder no matter where you go, but named organizations at least tend to draw better coaches or attempt to.
Can you get your softball goals and dreams accomplished without a "name" organization? Most definitely. But having a few more open doors to help short cut the process a little is always good. Being at the right place, at the right time, with other very talented players and teams is always a good thing. Having coaches and team organizational leaders that beat on doors and have tons of contact with college coaches is a huge plus. You still have to do a very large portion of the work yourself as a family, the organization (and the coaches especially) just needs to be there to help put your kid in a position to be successful and seen, she still has to put in the work and have the talent to get it done.
Better than being the best player on a bad team, in an organization that is not known, in venues that have no exposure, vs teams who also draw no exposure, offer little competition, keeps a few doors closed. Its a little common sense.
Go to where the Beverly Bandits, Outlaws, Lasers, <fill in many other well known org names>, etc all hang out and you will most definitely see a different level of competition and talent vs where the DQ Darlings, Bob's Waterbed Emporium All-Stars, <fill in many org's I don't want to offend>, play.
My two cents.