The U.S. has taken for granted that the trans-Atlantic alliance can survive these problems. It is now facing the fact that no agreements on any mission can be easily achieved. In focusing on keeping the British in the EU, the U.S. is currying favor with the union's most intense supporters. It is an interesting strategy. Don't recruit the closest ally, but countries that regard the United States with views ranging from indifference to disdain. I doubt it will work. I also doubt that a possible British exit will spell the decline of NATO. I believe that decline is already past the point of no return. But the United States is not without resources, and Washington has decided that protecting NATO is worth a shot. There are allies on both sides of the Atlantic, but they are not all EU members. The coalition shifts with every issue. This is not unmanageable, unless it is inserting deep divisions into the NATO structure.
The American fear is that the trans-Atlantic relations that define American strategy will collapse under European pressure. The allies' interests remain the same and therefore the relationship stays intact. It simply has to have a more dynamic structure that can effectively cope with European divergences.