Dear Mr. Gift:
[sarcasm] Which weighs more, a pound of saw dust, a pound of sand, or a pound of kitty litter? [/sarcasm]
When considering safety factors, cost should be no object. The question you should be asking if you love your family is “Which one works best?” not “Is saw dust good enough?” or “Can I afford NOT to do this?” instead of “How can I sacrifice my family’s safety to save some money?”
I think you are being penny wise and pound foolish about winter tires. Yes, the cash outlay would be high, but you can only mount one set of tires at a time, so while you use your winter tires, your summer tires sit in storage without receiving any wear. This means your tire cost should be spread out as a per-mile cost of operating the vehicle. The price of the new rims would be the only cost increase. The tires themselves would not significantly increase your per mile cost to operate your vehicle since both sets of tires would last longer than they would otherwise.