First I want to ask if any one else noticed the "firing tubes" they were shooting the rifles through, while the video may give a distorted perception they seemed to lower noise quite a bit.
Now I will take a shot at addressing FQ's question.
First, it is misleading to group America with the countries of the British commonwealth. While we share a language, culture and ancestors with them, our nation was founded, and our greatness achieved do to a completely different mindset.
They began as "Loyal Subjects", of the Crown, they obey and are RULED by their government, .
We on the other rule ourselves individually and and demand obedience to our laws of those we elect to administer the functions of government.
As to whether the Swiss type of military service would work here, yes and no. America's greatest armies were raised from the rural populations, Alvin York and Audie Murphy were both country boys, so were most of the troops of all our previous wars, their marching and discipline may not have been up to standards all the time, but men who grew up hunting to eat beat every army they ever faced, as an example, during the French and Indian war, the only victory gained by Britian's "European style" Army was Quebec, all the other notable victories were won by "Colonials", These same colonials, or their sons, were the men behind wood piles, stone walls, and trees that drove the best infantry in Europe back to and out of Boston.
Obviously the potential is there.
The problems come from 2 things, the mindset of today's so called citizen, and the way the force is employed .
Let's look at mindset first. We now have a mostly Urban population, dominated by intellectuals and a media that opposes the concept of Patriotism, governed by people you would not allow on your property. The concept of DUTY is either suppressed or redirected into feel good public works programs, every one is a victim, no one is responsible for anything, and every one is "entitled". The Officer Corps, once the source of inspiration to generations, have, int the Command levels, been reduced to Politically correct "managers" and bean counters, George Patton, or Chesty Puller would be court martialed in today's climate rather than celebrated.
My second point also goes to Mindset, the Swiss Army is a defensive force, it's duty is to protect the borders of Switzerland, period. They are a Neutral country and if you try to drag them into your war they will kill you.
America does not operate that way, first off we don't defend our borders, we just put the Border Patrol out to make sure no one hurts themselves on the fence.
Secondly, our Army is not constituted as a defensive force, it is, by law, a diplomatic tool meant to perform as instructed by Congress and the President. That type of employment requires a full time professional Army and can not be accomplished with "part timers" under the leadership of politicians who's support depends more on party politics than on the good of the country. (How hard would YOU fight to protect Obama and Pelosi ? )
To finish up, this type of force COULD work if its duty were to protect America, as opposed to "American Interests", but it would require the recognition of the concepts of personal responsibility, and Patriotic Duty.
As a side note I have to point out to FQ that a REAL "Libertarian", in the sense of the Founding Fathers, would dislike anything BUT a conscript Army that is only activated when needed. If you read ALL the Federalist Papers you would know that men like Jefferson felt that a professional standing Army in peace time was nothing less than a tool of government oppression. The only reason a small one was allowed after the war of 1812 was because of the drubbing we took from the British and Governors resented having to send Militia units to protect the Western Frontier, it is only since WWII that we have had a peace time Army of more than 50,000 men.