This already exists COTS
The Tesla PowerWall is the most high visibility version, but there are competitors, and you could even knock one together out of golf cart batteries and inverters. What Tesla brings to the table is positively exquisite battery-management tech, which assures the PowerWall battery will have a very long life.
"But I want a 5-minute solution, this is overkill!"
Actually no, it's a game-changer. Because now, the generator doesn't need to carry your house. Just replenish the PowerWall. So you can get it done with a Honda instead of a CAT.
If you go pure generator, then the generator must have enough capacity for your house's peak loads. You could end up needing a 20kw generator just so it doesn't croak when you put on everything at the same time, the 1 minute in 1000 that you do that.
The difference in generator size may pay for the PowerWall, especially since the generator does not need an ATS and other associated equipment; it only needs to be a generator.
A homes power use is actually quite small, and averages barely 1kw. That's because, while your dryer, water heater, A/C and oven may be going gangbusters on a summer evening, your house also has long periods of time of virtually no loads at all, as litle as 60W for refrigerator, night lights and your vampire loads (looking at you, Comcast). To run 1 KWH the whole month costs about $100 on your electric bill, so you can look at your bill and see for yourself.
I read this negative review from one of those anti-green naysayers, and he said "if your house draws 5kw on average, you'll need 70 PowerWalls for a 30-day outage". Ok, first, 5kw means a $500 electric bill, not likely. And second, a 30-day outage is insane, but if it existed, it'd actually an ideal use case for a small generator to complement a single PowerWall, just as I say.
Off-grid power is 10% generation, 90% conservation
However, only a fool would tarry forth into a power outage and keep using electricity like nothing had happened. Of course you're going to defer those wash loads. Whip up something on the stovetop instead of bake.
Understanding what your big loads are and avoiding heavy appliance usage is instrumental to weathering an outage.
Another option, on the electrical side, is enforce this with multiple panels. Have your main panel for utility-only loads, the ones you never want to run on PowerWall. Sauna, dryer, etc. pThen you have a subpanel for the loads that need to be on generator or PowerWall, and these are the ones that get automatically picked up on-the-fly.
You could even have a third panelboard full of loads that could go either way, like water heater or A/C. That panel gets a $30 Siemens or QO generator interlock, and is normally switched to "Utility" so it dies when the power fails. Once you are sure you have abundant capacity to drive one of those loads, you shut them all off, cut the panel over from utility to PowerWall, and switch on the load you want to run.
Of course, the best way is to make judicious choices about appliances, so you aren't creating a huge electrical load in the first place. For instance instead of baseboard heat, use a heat pump designed to effective at very low-temperature range, so electric emergency heat never runs. My house has an Empire furnace, that is gas and runs without any electricity at all. We also have a gas water heater; being on-demand it needs a small amount of power for controls. A PowerWall is in fact overkill for my house.