> he actually did claim that gradually increasing exposure does make you better at cold.
This claim is true in the broad sense, but not just from gradual exposure. In the Marine Corps, I went through quite a bit of training in dealing with cold conditions. There are specific training requirements for cold exposure, and a tried and true methodology for increasing a person's capacity for dealing with the cold.
Biologically, your body will naturally adjust to a climate over time. This occurs when living in warmer or cooler conditions for a period of time. The main controls are metabolism speed, sweat threshold, and activity level.
One of the key ways to quickly adjust to a new climate is by a forced activity level in it. When we'd train in a new climate, we'd always start by doing forced marches during the peak condition (hottest or coldest time of day, depending on which condition we were unused to) for a day or two, then move on to runs, then to actual mission based training. On the first day or two, we'd all be shivering, miserable, joints swollen, etc... By day 3 or 4, that would all go away. You're still uncomfortable, but your body has adjusted.
Once you leave that climate, your resistance to the extreme temperature slowly fades (as you're adjusting to a new climate), but it seems to me that I still carry a bit of resistance to the extreme.
We'd do refreshers whenever we got the chance, taking people on "polar bear runs" in the middle of the night in the desert, in shorts and t-shirts, and that would help sustain the resistance to the cold.