I understand from the standard teleportation protocol that 1 ebit is used up in teleporting 1 qubit and thus, cannot be used again -- and thus, we need 1 fresh ebit of shared entanglement between Alice and Bob per each qubit that Alice wants to teleport to Bob. Furthermore, the teleported state could have a phase-flip or a bit-flip error and thus, the $2$ bit syndrome of Alice's measurements need to be sent to Bob for Bob to recover the correct qubit.
However, is there a general proof of this statement apart from the heuristic argument? If so, is there a fundamental no-go theorem of quantum mechanics behind this restriction? For example, the fact that you need $2$ classical bits to be sent in order to teleport $1$ qubit appears to be less than optimal, i.e., even if it were possible to teleport a qubit using only $1$ classical bit (along with the $1$ ebit, of course), the no-communication theorem wouldn't be violated because one can only extract $1$ bit of classical information from $1$ qubit. Is there some other basic tenet of quantum mechanics that would be violated if this were to be possible?