It’s nearly impossible to be a barn rat at a modern professional barn. No one has the time (or wants to take it) to show a kid how to do anything, or let them try. My daughter would LOVE to be a barn rat and has offered to help with various tasks (dumping feed, grooming school horses, setting jumps) and is basically told to stay out of the way. I’m careful to only let her ask at seemingly appropriate times, and even then the answer is usually no. I think it’s a combination of professionals (including grooms) not feeling like they have the time, and the culture of not wanting a full board kid to be seen doing the grunt work.
I get it from a business prospective and respect the barn’s unwritten rules, but on the other hand when the professionals then complain that kids don’t know anything it drives me nuts. I put my foot down and make my kid do her own grooming and tack cleaning (out of the way of the grooms work area), but if she didn’t have a horsey mom she wouldn’t know how to do that either.
Kids can’t “want it” completely, but there isn’t much of an avenue for them to learn enough to be useful in a way that would help a trainer give them a chance as they get older.
FWIW the vets and farriers tend to be kinder to a curious kid- she sits and watches quietly and they usually take a min to explain to her what they are doing or let her take a closer look.