What, if anything, would you have recommended for the Kim family that is not in my top thirty?
Having some "extra" on a car trip is somewhat different than packing sensible for a hiking trip... For a drive through winter mountains, how about loads of warm clothes, sleeping bags, lots of food and a stove? Clothing, food and possibly a stove (if that's how you plan to treat water) is on your list, just bring more of it.
Actually, this tragic story demonstrates that the family actually fared pretty well
(relatively - I'm sure they felt utterly miserable, but they were alive and unhurt), except for the one who had to go on a poorly planned and executed rescue mission.
More about the clothing: If you absolutely insist on leaving the road to plow through unknown amounts of snow in unknown territory
(short answer: Don't!), then at least choose clothing appropriate for bashing in the snow. Gaiters plus clothes that won't soak up immediately and are at least wind proof, and good boots. Membrane clothing (such as goretex) are excellent for wading through snow, while your typical casual wear are possibly the worst. But in essence, this point is all covered by the term
clothing appropriate for the season.
For me, this story shows me that the urge to DO SOMETHING can kill you. Your average male will do whatever he can and spare nothing to save his family. Sometimes, the best thing is just to sit tight and essentially do nothing. Of course, while you do nothing you do your best to implement the thousands of small tricks that can make life just a tiny bit easier. Such as a more comfortable cushion, a bit more insulation, a new way of melting water...
Which brings us to the most important part of what you bring with you: Your brain. The mindset summarized in the STOP acronym and the British equivalent
make yourself some tea are extremely important.