Here's my understanding of how the system works. Note that I've skipped the solar in the diagram, since it's mostly a passive component of the system. I think it helps to understand the Powerwall system as a collection of components working together. In particular the gateway has three important components: the transfer switch/relay, the Neurio monitor to measure power flow and the controller itself that sends the commands to the Powerwalls to control their operation.
When the grid goes down, the transfer switch is opened and the Powerwall switches to islanding mode. In this mode, it just tries to maintain power at a certain frequency and voltage. I'm not clear on whether the controller is involved at all in this mode. It may just be the Powerwall itself that decides to raise frequency to curtail solar production, for example. I'm assuming the controller can program certain parameters of the operation, like the maximum frequency.
When the grid is up, the controller controls the operation to implement things like the time-based control modes. I'm assuming it can request that the Powerwall charge or discharge at any speed.
If the Powerwall itself fails, it shouldn't impact power to the house. Likewise, if the gateway controller fails, power should still flow from the grid. Obviously if the transfer switch gets stuck open, the house will have no power, but it seems to me that that would be unlikely to happen.
I'm not an expert, but I think this model describes the behavior that I've observed and have heard described on this board. Note that I believe that the connection between load panel and Powerwall is probably physically in the gateway box, but from a functional point of view, that doesn't change the way everything is connected.