Yup, even better if you power a separate battery charger with the generator, but let loads be handled by the battery+inverter. You don’t need to provision the generator for peak loads (which can be quite high), or even worry about high harmonic distortion from a cheap generator screwing with your electronics. It basically turns your system into a fully ‘online’ UPS.

Also, really handy for diesel gens which will ‘wet stack’ if you only run them for short periods or with light loads.

Provision a generator size which you expect to run say 6-10 hours at a time (at say 70-80% rated load) to fully charge your batteries, and a battery bank sized for 3-4 days of load.

Keeps your total gen hours much lower, much more efficient fuel wise (you can set the ideal load using your charger, and it will keep it that way regardless of increasing/decreasing loads), doesn’t require as big a generator if you have things like AC compressors, heat pumps, etc. with high start loads, and gives you a lot of redundancy and flexibility. It even gives you time to fix your gen or find/beg/borrow another if yours breaks, and you won’t need to keep the gen running overnight if you plan appropriately, so your neighbors will like you more.

Best of all worlds. Especially if you have natural gas, or get a tri-fuel generator and keep propane handy. Propane doesn’t go bad like gas or diesel can.