The US Army did want the M-14 and nothing else -- and I know the rest of the story of the rifle boondoggles from pre M-14 through the M16/A2 -- but the Armalite AR-10 failed on it's own. To look at the Fairchild/Armalite video from 1957, you'd think the AR-10 was everything a soldier could ever want, but in reality it had all sorts of problems.
The revolutionary multi-piece muzzle brake that looked like a tubular spaghetti collander was the part responsible for barrel blowups, along with ultra-lightweight Stellite barrels. The huge brake was the part that kept the AR-10 from bucking like a bronco on F/A, but when they removed it and replaced the hi-tech barrel with one made of iron, it experienced all kinds of jams, was front-heavy as sin, weighed more, and bucked worse than an M-14 on F/A.
The AR-10s ultra high-tech design was actually a liability, and the Army officers in charge of selection weren't about to get caught up in the late 1950s 'Space-Age' craze after just so recently fighting off human waves of Communist Chinese in Korea. They wanted a variation on something that they KNEW worked -- an accurized M-1 Garand. There have been grumblings about set-ups and rigged competitions, and though some of it must be true, it's not the whole story about why the AR-10 wasn't chosen.
Only the Dutch, Sudanese, Portuguese, and Tunisians adopted the Armalite AR-10, and I think that every one of those nations got rid of it after only a few years. The Dutch didn't even keep it for one year, I don't think.
I don't know by my own experience, but I hear that even the new Armalite civilian AR-10s are "Bang, Bang, JAM" in operation.