Last time I check the weightlifters and powerlifters weren't hurling themselves down a mountain while doing their movements.
Yes and no. Yes, the 1 to 2 minutes on/ recover defines what is typically an anaerobic based situation, but how many are really taking it to that level and digging that deep into their system to make it a true anaerobic situation. Not many, I'm guessing. For the average non-competitive athlete it is more like within that 2 minute section there is maybe 30 seconds of true anaerobic work. If your run lasts beyond 2 minutes you'll probably benefit from some type of endurance training that is anaerobic threshold based like running, cycling, inline skating. Spending the day on the mountain making multiple runs will benefit from short to medium distance endurance training.
I was never on a national ski or snowboard team, but did ski competitively and speedskated at both long track sprint and distance events. I have trained with the national and world cup teams for both long track and short track both on ice and off, and I assure you there is a lot of endurance training work going on, alot of running, alot of cycling. They just don't necessarily do it as their focus, and they don't do it all season long, they cycle their training with periods that focus on different elements such as alactic power, lactic power training, muscular, etc.
I think this is the heart of the matter. Competitive athletes train for basically three things, to have the fitness to carry them through their workouts, recovery, and most importantly for the competitive event. All that explosive training that skiers, snowboards, etc. do is for the 1 to 2 minutes that they are fighting for that first place spot. On race day, they are warming up, waiting, competing, lather rinse repeat.
If you are doing something to get yourself ready to make it through the day on the mountain, have fun doing it, and reducing your potential for injury and next day soreness, you are on the right track.
Since I put my .02 cents in, I will add what I am currently doing and why. Over the spring and summer I was doing a lot of cycling (road and mountain bike) - 5 days a week, some core - 3x per week, and some balance with strength building (ie squats on stability board type of stuff) - 3x per week. I was riding mostly to drop weight and improve my endurance, but I wasn't burning enough calories in an efficient manner to benefit from any weight loss. I work some long work days and with family stuff finding an 1 1/2 hours to fit a workout in can be tough. So I switched over from cycling to running for an hour 4x to 5x per week, some of it barefoot, and that has worked. I also switched up some of my nutritional habits and started wearing a body bugg so i can keep a close eye on whether I am burning enough calories on a daily basis. One other benefit of running over cycling is that running lengthens your achilles, cycling shortens it. I've learned from past experience that a tight achilles tendon is not beneficial for snowboarding.