Most inverters don't work without connection to the grid. If you flip the fuse or meter box to go "off grid" the inverter will stop working that very same second.

You can however get expensive inverters than do work of the grid, but they are, as mentioned, expensive compared to a simple grid-tie inverter.

The “expensive” part isn’t necessarily true. Some grid-tie inverters have limited backup capability for very little cost. IIRC the SMA Sunny Boy series can supply 15A 120V using a dedicated outlet. SolarEdge’s StorEdge is their standard DC-coupled storage inverter, and it can supply 5kW of backup power.