Dave, it seems to me to be a matter of choice. Our generator starts off the house batteries. We have had that set up on 3 different rigs. I have thought of separating them but, since it has worked, I have not been that motivated to do so. Our first motorhome had one house battery and a converter, no inverter. That battery worked for the small DC loads and could even run the furnace all night if needed. I do not think that it was really optimal though. We now have three large house batteries and an inverter, and they work fine even when dry camping (though it is really getting to be time to replace them), but we are not taxing them much. There has always been plenty of power left in the morning to start the generator.
It all comes down to your use and how much of a safety factor you want. Of course, make sure that you have good battery chargers that do not damage the batteries and test things periodically. We once ruined a bank of 4 AGM batteries with a charger that, although I thought functioned well, did not.