Because of political reasons. NATO honestly didn’t want a full-power rifle cartridge, and for good reason. Like the Soviets, they were looking at the German StG-44 and seeing a much more practical general-purpose battlefield weapon than the full power rifles that had been common up until that point. You didn’t need a rifle that was able to shoot farther than about 500 yards; data from the war showed that the overwhelming majority of firefights occurred well within 200 yards, and that volume of fire was a deciding factor in them. Therefore, these newfangled assault rifles and their intermediate cartridges made a ton of sense: these smaller cartridges produced less recoil, and less recoil meant less muzzle rise, less disorientation of the shooter, greater situational awareness, quicker, easier follow-up shots and more controllability in fully-automatic fire. So most nations quickly set to work designing their own assault rifles based around their own intermediate cartridges when the war ended. But the formation of NATO and the NATO standardization agreements forced the countries that had formed NATO to rework their plans slightly, as, so it was planned, they would have to eventually adopt one universal pattern of infantry rifle and one universal infantry cartridge to go with it.
Now, while the Europeans wanted a true assault rifle designed around a true intermediate cartridge, the Americans weren’t so keen on this. The US Army Ordnance Corps was a big fan of tradition, and such a dynamic, radical shift in firearms design did not sit well with them. While of course there were those in the US military community who did not share these feelings (the M1 Garand was in fact originally conceived in what could potentially be considered a prototypical intermediate caliber, .276 Pedersen, back in the 1930s), the people in charge by and large wanted to stick with the tried-and-true .30–06 Springfield. The Europeans were not budging on their desire for an intermediate caliber, however, so the US designed the 7.62x51mm cartridge, which, since it used what were at the time some of the most modern propellants available, achieved virtually identical ballistic performance to the .30–06. The Americans then tried to pass off their new 7.62x51mm cartridge as a genuine intermediate caliber, which didn’t really work.
In the end, the competition for the NATO cartridge adoption came down to the 7.62x51mm cartridge, which only really stuck around because the US refused to let it die, and the .280 British, a much better cartridge developed, obviously, by the UK to go along with their much less impressive bullpup EM-2 rifle. Ultimately, the US made a bargain with the Europeans: that if NATO adopted the 7.62x51mm cartridge, the US would adopt the Belgian FN FAL rifle. The Europeans reluctantly agreed.
Of course, as we know today, the US didn’t hold to their end of the bargain. The US Army Ordnance Corps, again blinded by tradition, went ahead and adopted the rifle they designed for the NATO trials, a select-fire, box magazine fed, 7.62x51mm adaptation of the M1 Garand which became the M14. Unfortunately, the M14 wasn’t half the rifle that either the M1 Garand or the FN FAL was, being overly heavy for its era and horrendously uncontrollable in full-auto, among other issues. The M14 would go on to have the shortest service life of any infantry service rifle in US history, as it proved to be just about the worst possible weapon imaginable for the Vietnam conflict that began shortly after it entered into service, prompting the introduction of the M16, a process which would be a fatal blunder for the Army Ordnance Corps and those involved with them.
In 1961 the Air Force, needing a weapon to replace their WW2-era M1 Carbines, adopted the Armalite AR-15, a scaled down version of the 7.62x51mm Armalite AR-10 that had been submitted to compete against the M14 to become the US military’s new rifle several years earlier. The Air Force, exceptionally pleased by the new rifle, then began to demand that the Army Ordnance Corps evaluate the weapon for the rest of the military, but unsurprisingly, the Ordnance Corps refused. Later on, the US began sending AR-15s to their southeast asian allies in the Vietnam conflict, as the AR-15, constructed primarily out of lightweight aluminum rather than traditional wood or steel, was lighter and easier for the smaller-statured peoples of this region of the world to carry. This would lead to US special operations forces, who worked closely with these allied forces, getting their hands on these rifles, and they quickly took a liking to them, finding them ideally suited to the jungle warfare of the conflict. Meanwhile, the M14′s problems were becoming such a concern that Secretary of Defense Robert McNamara ordered that production of the M14 be halted and that the AR-15 be evaluated and made ready to enter service. But the Army Material Command (USAMC), which replaced the Ordnance Corps from 1962 to 1985, had other plans. The AR-15 did indeed enter service, but without cleaning kits, with barrels that lacked chrome plated bores or chambers, and with corrosive ammunition that was too hot for the weapon’s action. The USAMC's plan was to make the new M16E1 rifle (E1 because it had not yet formally entered service; the M16A1 would come later, and even then, that was just the Army version with the forward assist; the Marine Corps, Navy and Air Force versions were simply known as the M16 and omitted the forward assist) fail even worse than the M14 so that the military would abandon it, and the results of their efforts were case heads torn off from excessive chamber pressure and bores plugged with carbon fouling. However, instead of simply taking the M16 back out of service, a Congressional committee was formed to investigate the issues. Eventually, the source of the issues was traced back to the USAMC, and they were found to be guilty of criminally negligent behavior, having purposely botched the introduction of the rifle in such a way that it cost numerous American soldiers their lives. The Springfield Armory has since been shut down, and the US military no longer relies on a state armory to produce its weaponry, instead contracting private companies to produce weapons for it, inspiring innovation through competition.