My test is simply to run aplay or speaker-test, then quit and run it again. The first run is audible, the second and subsequent runs are not audible until I provide a manual DMA event trigger.
When the ASP transmitter is taken out of reset it sends XEVT to the EDMA where it is "captured" in event register (ER). If that event gets cleared out before turning on the EDMA you'll end up with a scenario like you described, i.e. once you kick off a transfer manually it will then run. I haven't looked at the code you're using, but it sounds to me like the (re-)initialization sequence isn't correct.
The best way for the code to work would be for the EDMA to be configured BEFORE the McBSP. That way, even if the EDMA initialization procedure clears the ER bits it won't matter because when the McBSP is subsequently released from reset it will send a new XEVT to the EDMA.
Brad