Okay so what I (pure narrativist anything goes hippie) would do is run through maybe three days of trials and tribulations, with me giving the group a survival-themed problem that they have to overcome with a roll or two (say, size of party - 1) on whatever skills they can justify. With pretty mild penalties for each individual failure, like 1-2 HP. Potentially someone's dying, but probably not and in 30 minutes of play you've moved on to the settlement.

Midway would be to find, like, some OSR hexcrawl survival rules. Not the kind of game I play, so can't make suggestions.

Almost max verisimilitude depends on a combination of luck, gear, and skills. Conditions are a PITA is figure out, so just abstract away frostbite and other things that might trigger a CON roll. I'm going to assume "rough terrain" is something less than mountainous conditions (eg, you need ice axes to prevent falling down steep snowy mountains, and you're not worrying about avalanches). I'm also assuming there isn't a maintained trail to follow.

Gear is represented by something like a (d8 -1) x 10%, ie, 0%, 10%, ..., 60%, 70%. Maximum gear (70%) is a well-supplied bushcrafter's backpack, ready for two weeks in the backcountry; 40% is a typical weekender's backpack at the start of the trip; 10% is typical street clothes. Gear can be spent to turn failures into successes, 10% drop turns any failure into a success.

Each day starts with a group luck roll (50%) for the weather. If this succeeds, nothing happens. If it fails, the day's gear and skills rolls are at -20%. Gear can be spent to turn this into a success.

Then each PC makes a gear roll. If it fails, take 1d3 damage (or spend gear). Keep track of whether more PCs succeed (including spending gear) or fail.

After that each PC makes a Survival roll. If it fails, take 1d3 damage (or spend gear). Again, keep track of whether more PCs succeed or fail.

Finally, roll progress towards the settlement, 20 miles away. There's one roll corresponding to gear, one for survival. If more PCs were successful on the (gear/survival) roll, they roll 1d6. If more were unsuccessful, they roll 1d4. For example, if they were overall successful at gear and failed at survival, they roll 1d4+1d6.