Why would your battery be drained? Leading up to an outage, you would be using utility power to run your heating and other stuff, not the battery. The battery would only need enough juice to stay charged at full. And because of the way modern battery packs work, that means that it would only charge up at periodic intervals, never letting the capacity get below maybe 80% to 85%. (But could be set arbitrarily high, depending on need given outage frequency and average length.) And that charging would happen during non-peak hours, so there's no extra load on the grid. I have an EV, and that's how I have my car set up today to charge; in the middle of the night when the grid is nowhere near peak load. This "problem" has been solved long ago.
The availability of solar or wind power has nothing to do with this conversation. We were talking about load distribution and how much the grid would be taxed with batteries at people's houses, not energy sources. But just to counter the FUD...
Wind turbines work fine during bad weather if they're properly maintained. Plenty of countries with more bad weather than we have generate plenty of wind and solar power, including places like Sweden, Finland, Denmark, Norway, and Germany.
What you're likely thinking of is the problems that Texas had in 2021 due to a freak ice storm. A bunch of right-wing nitwits got all over television and the internet to decry how unreliable wind power is, when in fact, it was natural gas that took the heaviest hit in failing to supply enough power. Windmills actually worked much more reliably during that snowstorm. (Citation)
[W]ind turbines — like natural gas plants — can be “winterized,” or modified to operate during very low temperatures. Experts say that many of Texas’ power generators have not made those investments necessary to prevent disruptions to equipment since the state does not regularly experience extreme winter storms.
While Webber said all of Texas’ energy sources share blame for the power crisis, the natural gas industry is most notably producing significantly less power than normal.
“Gas is failing in the most spectacular fashion right now,” Webber said.
Dan Woodfin, a senior director at ERCOT, echoed that sentiment Tuesday.
“It appears that a lot of the generation that has gone offline today has been primarily due to issues on the natural gas system,” he said during a Tuesday call with reporters.
As for solar, it can be impacted, but it's rather unusual for it to be. It has to be really cold or a really heavy snowfall, because the surface of solar panels generally are warmer than the surrounding area, meaning that snow generally melts off of it quickly. Also, solar panels are typically mounted at an angle, which induces snow to slide off of them. And even panels that are partially obstructed generate decent amounts of power. (Citation) Again, in practical use, this is a non-issue.