I can't imagine that cutting up potatoes will get rid of the nutrients in the skin.

Well, in that initial map scroll-in, you can clearly see that the video is set somewhere in Oslo. (Or at least gives the impression.)

Keeping a lid on help keep the temperature more consistent, as there is much less heat escape and evap into the atmosphere.

Water boils at 100C, but did you know that steam exists at temperature higher than waters boiling point? So steam above the water can go upto 120C depending on the weight of your lid. So when you put a lid on the pan you are semi boiling/ semi steaming which is why it cooks faster.

Actually, putting a lid on will give an incredible difference if you have a pressure cooker. You can heat up water to 120 degrees Celsius because of the pressure inside, and that should make those potatoes done faster.

The water isn't heated when they start. It would get to a boil much quicker with a lid on.

Actually water starts to bubble around 70 C, and small bubbling vs heavy boiling makes a great difference in cooking time. But of course it doesn't matter whether you achieve this by turning up the heat or putting a lid on the pan.

Incidentally, just like when BBQ make sure you pierce the potato with a fork. Otherwise, yes you'll have cooked potato in <1 minute, however you'll still have to clean the microwave so it would be more like a time-shifting step rather than a time-saving step in the cooking process.