I think something like that system could work.
Maybe 4 hours, and it's more that each 1 unit could be a day's worth. So if you roll well, you've got enough wood to keep warm for 3 nights.
I wouldn't include stone unless you're specifically trying to build something. What I'm thinking is that most of the players would be the kind of people who travel often, or don't do their own construction, so I wouldn't want these checks to take all day. I'd also make finding game more interesting though. A check to see if you find something, DM decides (maybe rolls on a chart, or 10 is small game, 15 and 20 are larger?) to see what you found, and you take the shot. Either you kill it, wound it and it attacks or runs away but is easy to follow, or miss it and it attacks or escapes. Then you get however much meat and a fur.
I suppose if you want to play a game where you just settle down and try to build a farm, that's fine. I know there's appeal to that, and for anybody interested, there's a game called UnReal World which is a survival game in Iron Age Finland which is basically all of that. Farming, hunting, fishing, gathering berries and mushrooms, building various shelters, and panicking while trying to build a fire because the ice cracked and you barely made it back to shore.
I think of the survival aspect as something you do while investigating what's been killing seacattle and leaving eviscerated corpses on the shore, or while tracking down Flintless Dave, who kills people after they've set up camp so that he gets their valuables and a warm place to sleep. That and specific situations like when you notice a blizzard is coming in and you've gotta either weather it outdoors, requiring turning your sled into a shelter and having enough food and wood, or remember that you know a farmer or rest stop nearby and impose upon them to let your group stay there. Maybe they've got enough food, and hopefully everybody can keep from killing each other before the storm passes.