I know you very well might be right -- and I sure hope so.
A lot of that could depend on how the important people in her life respond. All kids at that age can be damaged if they react by saying it is important that she/he improve her skills, as if they somehow failed.
Kids generally know why they didn't make a team. If it's because a coach was partial, telling the kid to work harder isn't helpful. If it's because their skills weren't up to par, the kid knows it, most of the time. If they were on the bubble and there's no obvious reason why they weren't picked, there's nothing to say. So IME, it's better to empathize -- "oof that sucks, I know you tried your best, I'm sorry you're hurting" -- and let it go from there. It's okay to let kids wrestle with discomfort and decide what they want to do with it.
I will add that over the decades I've been familiar with school sports, I can say without risk of error that not all adults & coaches are objective when it comes to team sports.
I've been in the same position in multiple roles. There's nothing a person can say to remove the sting of not making a team, no matter what the reason, so acknowledging the hurt and hard work is really all you can do.
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