I think you read the sequence of events in the movie backwards, although the framing makes it seem like that's the implication. The movie opens with Dottie telling her boy that it's important for his brother to win every once in a while, which seems to foreshadow the end and indicates she dropped it on purpose. But that scene is actually in the "present," and chronologically takes place long after she dropped the ball. In the past, when the movie is set, Dottie *never* lets Kit win, at anything, her entire life. And Kit was actually a pretty impressive person in her own right, and should have had lots of accolades, except that she was constantly overshadowed. And going into that final at bat, she indicates she fully is ready to do it again. But when Kit actually pulls it off, and wins the game, Dottie sees how much it means to Kit, and realizes the importance of putting competition aside when your family. It wasn't until she saw Kit get a moment in the sun that she realizes she was destroying their relationship.
It's actually backwards: Her dropping the ball set in her mind that letting your sibling win every once and a while is a good thing, not the other way around.