OK, you're leaving out one major thing, but also a bunch of other things. I'll deal with the other things first, then get to the major thing at the end.
Assuming your numbers are right, and I think you need to provide your sources, that would be about $3,342.87 per person in the US. Of course, those would last a good 15 years or so. That means $222.86 per person per year, or $18.57 per person per month on their power bill. So, not too bad if your numbers are accurate.
There is the question, of course, of if your numbers are accurate. So, you have $1,170,000,000,000 worth of battery storage and let's assume a cost of something like $300K per MWh of storage. That means 3,900,000 MWh, or 3.9 TWh. That seems in line with average residential electricity usage for the US of about 4.2 TWh per day. Of course, that's the whole day. If we look for the place in the US with the longest night on the shortest day... well, we get places in Alaska that have 24 hour nights because they're in the Arctic circle. If we ignore the state that has about 740K people, the longest night is in NorthWest Angle, Minnesota and the daytime is still 8 hours and 14 minutes. On the same day, the place with the shortest night is Key West, Florida, which gets 10 hours and 36 minutes of daylight. Sure, it's not the best daylight for solar power in either of those places, but you're still being misleading about how much power needs to be stored.
The reality is that, if built so that the mean power generation is exactly the mean power requirement, then the mean power generation in winter is roughly 40% of the overall mean. So, you don't have to actually store 100% of the needed power, reducing your number. Plus the fact that you don't actually have to build to provide exactly the mean power needed, you can overbuild so that you have more surplus power year round, and a higher percentage of daily required power in the Winter (you're sure to point out that it needs to be stored, but more on that later) Then there are other factors, like typically reduced power needs after daylight hours. Then there are households that have their own battery storage which can reduce the need for grid storage (and remember, with EVs everywhere and systems that let houses draw from their cars at night, and a smart grid, grid storage requirements could be even lower). Then there are the types of storage that most households have even without battery storage. For example, about 16% of residential electrical usage is for heating hot water, which is then stored in a tank that can keep it hot for days. So if you have a smart system that heats the water only when non-stored power is available, a percentage of that surplus capacity from overbuilding can be stored without needing battery storage on the grid. Not to mention that, for home heating, you can heat hot water during the day (with a heat pump system) and release it into the house at night to heat the house as well, reducing the need for electricity at night for heating.
Aside from that, remember that the US is four time zones wide. So with a reasonable power grid, and some overbuilt storage for solar power in the different time zones and in the regions further South with longer days, you can get a lot of extra power so that the overnight storage requirement is even less.
OK. Now I will address the major thing you left out that I mentioned at the start. That is that powering the entire US with just solar and batteries, while possibly a fun mental exercise, is contrived if being used to calculate the battery needs for the grid. Even ignoring other sources and storage systems like hydro and hydro storage, geothermal, etc. there's the obvious other power source: wind power. Wind is strongest in the US in Spring, but second strongest in Winter. Whatever the mean capacity factor for wind power, it will be above that capacity factor in Winter, reducing the needed storage for overnight by a lot.
Ultimately, when you consider all of the factors, you clearly would not need 100% of the days energy stored for Winter. Realistically, with a well designed system, you would need a third or less. So that portion of the average utility bill would be around $6 per month per person or less. So, not actually bad at all, really.
Now, it is worth noting that we are only talking about Residential usage. Other uses like industrial and commercial are about twice residential usage. Still, most of the same factors apply. Plus, for most commercial usage, the drop in usage at night is a pretty big drop. Industrial varies. There are a lot of industrial customers who also stop at night, but there are plenty of big power draws that can run all night. Overall though, given most of the same arguments that apply to residential, commercial and industrial would also only need about a third of a days power usage or less stored for overnight usage.
There is also the drop in price that goes along with scale. I cited $300k per MWh, but it's already down to around $180K in some places. With battery prices dropping and other battery chemistries and perfection of the methods for building out this kind of storage, it should get even cheaper. So, the quoted price drops by potentially another 40% or more.
In the end, it seems like you think you're presenting a showstopping giant number, but it seems like you're actually presenting a great deal. The current US administration just asked for $1.5 trillion for the military. Not for decades like the grid storage, mind you. For a year. Also, while all sorts of arguments can be made about defense of the country, etc. the basic reality is that the money is for oil related wars. Big numbers don't really look so big when you look at the other big numbers we're already spending.