It was my impression that her abandoning a pure Vulcan approach was the point of the prologue.
Huh, how do you mean? Do you mean that Burnham thought the "logical" thing to do was to try and neck-pinch her captain and take command?
It was pretty clear at that point she was absolutely terrified, thus not thinking logically at all. And (unfortunately) the way the story is structured, it implies if they did give a "Vulcan hello" the war would have been averted.