Earlier quoted context omitted.
> you need three emails sent in rapid succession This is partially correct, but they do not need to be in rapid succession, and therefore Bob does not need to be online.
Alice's ephemeral private key must be kept as long as the whole handshake. Bob's is a bit shorter (between the last two messages). If the messages are slow to come, those ephemeral keys become less and less ephemeral, and could actually be stolen.
> If your security model relies on this difference then your security model is broken. It’s worth noting that Signal does NOT rely on this difference. It relies on participants being mostly online to permit frequent rekeys and not having to retain old keys indefinitely.
Signal does not depend on TCP being "bidirectional" as lvh said, it depends on participants being mostly online. This has nothing to do with the transport properties of e-mail vs. TCP.