I do it different than most here do. (not unusual) I run my generator all night, when the power use is greatest. (furnace and dehumidifier)
Before everyone gets their panties in a bunch.... I do NOT use campgrounds. The only ones I have to worry about is my camp.
With EU 2000 50' away, I cannot hear the generator with a Tv or furnace running. My unit has 3 TVs, so at least one is usually on.
After breakfast we turn the generator off, and coast thru the day on the batteries. Unless it is super cold out, when the furnace would be running... If it is that cold, then I need the generator anyway to power the heat tape I installed on all of the plumbing lines.
The OP is correct in that driving will NOT fully recharge the batteries... At least not without some modifications.
A generator is the most flexible way to meet his power needs... All of the rest of it, solar, inverters etc, are nice to haves in ADDITION to a generator. But inverters drain batteries and solar doesn't work at night or when it is raining. Bottom line: Get a generator first.
As for the brands.... Lots of debates out this.
The more you use one, the more likely you are to end up with a Honda EU2000-2200.
At the dog trials I have been attending since 2001, I have seen a lot of campers using generators. The overwhelming choice is Honda. Those that show up with another brand, end up with a Honda in short order.
Some things just flat out work.