Yes, but there are nuances. As I said, the switchover causes problems. And once the power is interrupted, even briefly, my spa starts up at full blast 5 minutes later. If I'm not home, there can be an endless series of AC start attempts and Powerwall recycles until the house cools off at night. I can turn off my AC remotely using an app but I have to know a power outage occurred and my AC can't be in a reset cycle which takes about 15 minutes to complete. Once the reset cycle is done it attempts to restart if the temperature demands.
If I'm home I can do load management but it isn't easy. If the Powerwalls reset the solar is commanded off for 5 minutes mimum. My solar isn't commanded on until my Powerwalls reach about 92% SoC. If my house isn't using much energy (which it isn't if the AC isn't running) the Powerwalls recharge really fast and command the solar back off. As I mentioned, my HVAC is a computerized system. I can tell it that I want to cool but it does a system checkout and commands the compressor on when it is damn well ready. This in itself can take 15 minutes depending on the last reset. If I can get it going it will continue to operate unless it reaches its set point or tries to change to high stage, but I can manage those. But it is a PITA.
Since Tesla was proposing removing the system completely I asked what would happen if I did nothing. At least I had some AC during an outage. They told me they weren't happy with the Powerwall reset cycles and may not warranty my system.