The Swiss are in a slightly different position. America (the U.S. of A. specifically) has no natural enemies. To the north is Canada, who wouldn't hurt a fly, and to the south is Mexico, who's been dealt with. It has nobody constantly trying to kick down its door, nobody who's wars are spilling over, nobody who makes claims to territory based on historical ownership and/or population majority.
America is safe. There's a zero-percent chance of a war erupting from outside of its borders that might affect its borders. The Swiss are not safe. They're surrounded by dozens of warring states that have only recently calmed down, with each one having held claim to some piece of land somewhere that they don't currently control.
Americans arm themselves to push their own agenda on the rest of the world. Its purpose is the projection of power, and the American military is built exactly for that. The Swiss (and most of the rest of the world) arm themselves to protect themselves.
Now, I don't necessarily disagree with making everybody essentially a part of a national militia. However, I'm not certain the consequences of such an act (e.g. no defense spending, no standing army, etc.) would happen the way you envision. I do agree that everyone should learn how to use and care for a firearm from a young age. It'd probably be more beneficial than learning how to drive at the very least.