I wouldn't go as far as exhaustion
I would. Heaps of precendent for it in the DMG:
Whenever the temperature is at or below 0 degrees Fahrenheit, a creature exposed to the cold must succeed on a DC 10 Constitution saving throw at the end of each hour or gain one level of exhaustion. Creatures with resistance or immunity to cold damage automatically succeed on the saving throw, as do creatures wearing cold weather gear (thick coats, gloves, and the like) and creatures naturally adapted to cold climates.
That's extreme cold (-18 degrees C) but the save is every hour.
There is also the rule in Xanathars where missing a nights sleep triggers a DC 10 Con save or suffer a level of fatigue.
I would happily rule that a night in the rain and mud/or and cold without shelter and/or a bedroll triggers a DC 10 Con save (either from the cold, or from near total lack of sleep); on a failure you suffer a level of Exhaustion. On a really cold night, I'd impose disadvantage on the save.
And seeing as you already need to sleep (barring elves) for 6 hours to gain the benefits of a long rest by RAW, a player that decided not to bring a sleeping bag and shelter isnt getting the benfits of a long rest due to not getting a solid 6 hours of sleep unless they make that Con save either.
Tents hold a few people, so they can be split up among the party (just like normal hikers do). A blanket will also suffice for a bedroll (you can roll your cloak up into a pillow) unless the temperature gets really cold.
I think you're right that there was some "gaming" of the system going on, which I think had to do with a lack of experience playing with the Encumbrance variant on the part of the players and facing the hard choices that entails with regard to what to include in their equipment.
If the question is even asked, those players need to be slapped down instantly to stop any future attempts at gaming the system.
Player: 'Hey DM; do we need a bedroll and tent when we're camping? I'm leaving mine at home.'
DM: 'What do
you think? (grins evilly) How about you try it and find out?'
Bringing tents and bedrolls may seem like a forgone conclusion to us,
Your players are 'us.' I dont really get the anachronistic bit either; people needed shelter in the dark ages just as much as they need it now.
I mean, only an idiot wanders off into the woods for a long hike with no shelter or sleeping bag, unless they're a survival guru (and even then they take a shelter, relying on survival skills n an emergency).