Why does the entangled pair need to be entangled to perform teleportation?

You can think of entanglement as a way to share the information between several qubits (this is definition of entanglement - the state of the system in which information is distributed across the whole system and can not be represented as states of individual subsystems).

• First Alice and Bob's parts of Bell pair are entangled.
• Then Alice's message qubit is entangled with Alice's part of Bell pair, so that information carried by it (the $alpha$ and $beta$ in the $alpha |0rangle + beta |1rangle$ state) is distributed across the whole system.
• Measuring two qubits in Alice's possession "squeezes" the information out of them and into the last part of the system, Bob's qubit.

Without the initial entanglement, Alice's qubits don't have a way to share information with Bob's qubit.

In more mathematical terms,

• If Alice's and Bob's qubits are not entangled initially, the state of the system can be represented as tensor product $|psi_Arangleotimes|psi_Brangle$.
• Adding message qubit corresponds to the state $|m_Arangle otimes |psi_Arangleotimes|psi_Brangle$
• Now, if you only perform gates on the first two qubits, you can always represent them as $G_Aotimes I_B$, and the state of Bob's qubit never changes.