I think it's just a cultural difference. The "two sheet" system is commonly used in homes in the US. So US hotels use the same system because it's familiar to most of their guests. (The US is a big country with lots of domestic travel, and most hotel guests would be Americans rather than travelers from abroad.)
(As a clarification, the most common system involves a sheet on the bed, and then one more sheet between the person and the blanket. You've found a bed that has one more sheet over the top, so it's really a "three sheet" system - the third one is not so common in people's homes. You could certainly remove the top one if you don't like it there, since it won't be relevant when you're sleeping under the covers.)
instead of using a proper duvet cover like most people do at home
That's not what most Americans do at home, in my experience (I am one). Even if there is a duvet with a cover, they will usually still have a sheet between the person and the duvet. And many people don't use duvets at all, preferring wool or synthetic blankets or a quilt.
What's the reason behind this?
As I said, I think it's mostly just the way things are. But it has a few benefits that I can see:
Duvet covers can be awkward to get on and off; it's easier to just change sheets which are not attached to the blanket or duvet.
A duvet cover is twice as much cloth to wash compared to a sheet.
A sheet usually extends over the edge of the bed, whereas the blanket usually just reaches the edge. This makes it less likely that you'll come completely uncovered, especially if there are two people in the bed. It's also easier to tuck the sheet under yourself to avoid drafts.
With a separate sheet, if you get too warm, you have the option to put the blanket aside and just sleep under a sheet.
Is it possible to ask the hotel to change it for the duration of my stay?
Of course you can ask, but my guess is that most American hotels will not have duvet covers on hand, so they wouldn't be able to accommodate such a request. You may also have a hard time getting them to understand what you mean, since the concept of sleeping directly under a duvet cover may be pretty foreign to them.
It's unpleasant because the two sheets are eventually untangled from the blanket
Well, I don't think everyone has that problem. Maybe you move around more than average when you sleep.