In short, I mean if they have bad experiences with the dad showing a kids stats since 9u and on...or they don't want to get involved with a parental conversation for a kid they are likely not into, or for the abundance of reasons listed above that they just would prefer not to talk to to people/parents.....logic would dictate they don't show up in 5 articles of clothing showing what school they are from....why not just dress like someone's Uncle Sal and pretend to just watch the game like a normal person rather than dressed in identifying garb?
I think, and no I do not have the complete answer to your question, there are several reasons. In some cases wearing their team attire adds legitimacy to what they are doing, for the players and coaches that do not know them. They are also at work and representing their program. I'm sure in some instances they get into some events for free or at a reduced cost because they are coaches, therefore they wear their attire. I'm certain that I'm missing a few more.
And by the way, in my old days I was a uniformed police officer driving around in a marked police car. And at times I would buy my lunch and try to sit somewhere off the main streets in an attempt to eat quietly. And it NEVER failed that during that time someone would walk or drive up to ask a question. All I was trying to do is eat quickly while monitoring the radio, in case I was needed. If I sat somewhere more visible I could see the people driving and walking by, gawking and muttering under their breath, assuming I was goofing off or whatever else they may have thought. And all I was doing was my job, while wearing my team's attire.