We have three young children and have been really happy keeping our old car, a 2008 manual-transmission Honda Fit. When the third child was born, I had to get different car seat to fit three-in-a-row. We spent about $200 to get a Chico KeyFit and two metal-framed Dionos, all slightly used from Craiglists or Nextdoor. We get great gas mileage, insurance is low, and can fit everything we need for all five of us for all of our adventures, such as all the gear needed for multi-day backpacking trips. The front wheel drive and manual shift would suffice for most snow driving. If the roads are much worse, do you really want to be driving anyway?
We have a bolt-on hitch that can hold a bike rack or tow a small trailer. I installed myself and probably cost $100 on Craigslist.
The Fit is also our only car, so we don't have options of driving two cars when family or friends need a ride. We bike a lot instead of driving. I think we'll make do with this setup for as long as we can, and then at some point get a used 8-seater van.
Remember, the more expensive, bigger car now isn't simply the purchase price, but also the higher costs for fuel, insurance and maintenance (esp. things like tires are much more expensive on SUVs and vans compared to a sub-compact like the Fit).