Bandwidth can be stored in this manner only for as long as qubits can be stored. At present, most quantum memories realized in experiments have a storage time of well under a second. Some last slightly longer, but with big tradeoffs. Stopping decoherence in quantum memories is hard enough that it will likely be a very long time before classical bandwidth becomes the bottleneck!

Though I appreciate the idea of 'storing bandwidth' by just sending lots of qubits ahead of time, that ignores the extra communication overhead needed of tracking them: When Bob receives a qubit from Alice, he needs to know which of his qubits it's paired with. That implies that Alice needs to send not just her qubit, but also some kind of identifier that will allow Bob to match up the received qubit to its proper Bell pair. That identifier, and any protocol overhead needed to keep these identifiers in sync between bob and alice, will consume bandwidth, and reduce the 2x gain.