This is false. I've tested this possibility by turning off phone and Signal Desktop still receives messages. Description of plan around which protocol was build is here [0].
That's slightly better but still cumbersome to set up and with the primary device as a SPOF. What does the recovery procedure look like if you lose the master?
From what I've read in docs posted above, identity key is copied from S-Android to S-Desktop. If you lose S-Android, there are two possibilities.
1) Your keys are safe (device was encrypted and/or you've wiped it remotely, whatever). In such situation you could be able to transfer keys from S-Desktop to new mobile. AFAIK there is no such functionality yet (remember, its Beta).
2) Your keys are not safe. In such case no recovery is possible. Notify all contacts about the fact that they should "reset secure session", forgetting your Signal identity and establish new Signal identity.
Anyway, this is the only solution on market with secure chats and multi-device sync.