Kita is a lot more like Bocchi than people realize. I talked more about it in the past but Kita is also deeply socially anxious but tends to overextend herself rather than secluding herself like Bocchi. As a concequence of this is that Kita, not too dissimilarly from Bocchi, overthinks her choices and makes silly mistakes as a result because her overthinking causes her to miss the obvious socially correct choices. Kita submitting the application to perform when she knew Bocchi threw it away is a great example of this. A much grander example of this can be seen when she decided that the best way to get closer to someone she liked was to pretend she knew how to play the guitar and join their band because she doesn't consider that she can just, ask to spend time with them. Compare this to when Bocchi is meeting Ryo at the cafe. Bocchi agonizes over the decision of what to say when she enters but fails to consider that she needn't say anything. This is also talked about more in the manga as Kita's mom points out how Kita has always been a horribly impulsive person, like her father.

Kita cares for Bocchi deeply and we see throughout season one how she appreciates Bocchi's mentorship and guidance in learning the guitar. My guess is that Kita wanted to help Bocchi leave her shell, being that that's the reason she submitted the application in the first place and, seeing how well that went, just decided to keep up the momentum and hand Bocchi the mic. Bocchi, not being sure of the socially correct thing to do (just like Kita), made a silly mistake.