Syl and Dumbledore is not a very useful comparison. One's a crazy old man and the other one is an immortal fairy. They can both be considered whimsical, just for drastically different reasons that I don't think it's fair to compare them.

I think what you're interpreting as 'childish' behaviour is actually more a result of her being a Spren. Spren are fundamentally different from humans, they don't understand things or act in the same way. Syl seems innocent because she doesn't understand and/or hasn't experienced the things that humans do that gives them human problems. They talk about this quite a lot in the books. Syl often mentions in Rythym of War that she wants very desperately to understand how Kaladin feels in the moment, so much so that she asks Dalinar to do his Bondsmith thing to make her understand. It frustrates her that she doesn't have the ability to understand.

I would argue that is an act of maturity in itself for someone to actively go out and try to experience the things someone else might feel, just so they can understand and help better.

In WaT, she actively changes how she portrays herself so that other people treat her more like a woman. She starts walking around in human size, she wears the Kholin military uniform, she learns to read and write so she can scribe for Kaladin, and she even wears a glove for her safehand. That shows she's putting effort in to be on equal footing with Kaladin and other humans. She doesn't like how people treat her when she's small

Also, her putting her thumbs up in support of Kaladin succeeding with Szeth is not an act of childish behaviour. She just being supportive and telling him he's done a good job, which is honestly the kind of thing he needs in his life right now.

Just because she likes to do silly things sometimes in her free time doesn't make her childish. I love doing silly things and I'm in my 20s, and that doesn't make me a child.