But can’t those quantum states themselves BE that information?
No, because the quantum information that is transmitted is has a built in degree of uncertainty. Performing an action on the entangled particle on one end will result in a somewhat random resulting change in the particle on the other end. Without the additional transmittal of classical information, you can’t know exactly what operation was performed on the other end.
So, essentially, quantum teleportation does not increase the speed of information transmittal at all. What it really does is allow you to tranmit a much larger amount of information at the same normal subluminal speed that you use to transmit the classical information. The quantum portion of the information is transmitted instantaneously, but until you get the classical information transmitted conventionally, it’s useless to you. Once you get the classical information, as small as 2 bits of information, then the entire transmittal, of classical + quantum information can be completed.