@KZO Just pretty much like what @PredatorDuck said, I think that's exactly the joke. People from distance are thinking that Tsubasa is boldly making move on Koyume by grabbing her arm, but Tsubasa is just scared and desperately holding onto Koyume in reality.

@KZO Sometimes Japanese is very non-specific. The original sentence is 翼ちゃんの方がそう来るのね which would literally translate to "Tsubasa's way came like that".
She also can't see that Tsubasa is scared, so from her perspective it looks like Tsubasa is taking the lead.

I'm pretty sure "initiative" is not the word the Matron used there. Tsubasa is supposed to be scared.