If your outage is longer than expected, the battery will recharge in a little over an hour and be ready to go again (if the grid comes on for a bit, or from your generator, a neighbor's etc).
Practical problems we had the one time we actually needed our generator were:
* (remember to turn off the main breaker)
* whoops, we blew the GFCI when we tried to back-feed power through an external outlet, so we had to run an extension cord through a door (with all that implies). This is the only thing that actually took us by surprise and took some debugging in the weeks following.
* the 120V generator only one half of the split phase, so every other circuit in the breaker box doesn't work. Pick your half carefully!
* the generator is strong enough to run normal appliances, but not the well pump (note: we had a completely different plan for heating). Here I suppose a battery could've been useful ... but then, the reason we didn't just buy a bigger generator in the first place was cost (for an extended-outage event that only happens once in multiple decades; for anything shorter you can just ... not open the fridge).
* (just a note that with the increasing electrification of cars, some variant of "plug your house into your car" is likely to become more of a thing)
While I've seen the grid flicker, this has only ever happened just at the start or just before it comes on permanently, so I'm not sure how useful it is to consider the "comes on for a bit" case either.