The short version is that the US wanted something that could penetrate helmets at ranges further than the lighter cartridges then under consideration were capable of.

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The 7.62 NATO cartridge was developed as the T65 cartridge in the late '50s. It was an outgrowth of advancements in powder that meant that similar performance to the 30-06 cartridge standard at the time - M2 Ball - could be achieved in a lower-volume case. The chief advantage here was the shorter case - shorter cases meant shorter actions and marginally lighter ammunition and loading devices. These sound insignificant, but on the scale of an army, it has the potential to be significant - rifles can be lighter and less expensive, and ammunition logistics becomes easier. Additionally, while this change was marginal from the perspective of the US Army, taking every other NATO army into consideration changes the dynamic. 7.62 NATO was a huge step up from the rimmed .303 British cartridge, and it offered a common cartridge among the myriad other calibers then in service that didn't favor any particular power by standardizing on what was already in production.

However, it's hard to understand the adoption without looking at the whole story - the NATO Light Rifle Trials. One of the first efforts of NATO standardization, the NATO Light Rifle trials were looking to develop a common rifle and cartridge for NATO forces that could serve in the infantry rifle, submachinegun, and light automatic rifle roles. Three rifles would emerge as contenders for this program - the American T44, the Belgian FN FAL, and the British EM-2. Contending for cartridges, the US put forward the T65 cartridge and the British put forward a pair of smaller-caliber cartridges known as .270 and .280 British.

While the US rifle was designed with the American T65 cartridge in mind, the competing rifles were not. The FAL was an outgrowth of parallel FN efforts on the full-sized-cartridge FN-49, and it saw iterations and developments in a number of lighter cartridges, including .30 carbine and 8x33 Kurz. Ultimately, both the Belgian and British designs would be put forward in using the British .280 cartridge for the purposes of the competition, and development took a distinctly focus than the US efforts. The US Army was effectively unwilling to consider anything with less performance than the M1 Garand's 30-06 cartridge, hence the near-identical performance of the T65 cartridge. Meanwhile, the British and Belgians were looking at combat experience during the War, where engagements at longer ranges were rare, and their tests of captured German 8x33 cartridges found it offered adequate performance at medium ranges and more controllable sustained fire. The .270 and .280 British cartridges were an outgrowth of that, adding a bit more power to reach out a little further than the ~300 yard effective range of 8mm Kurz, but using a lighter cartridge for a flatter trajectory.

Unfortunately, the diverging design philosophies would come to a head once the trials started. The US was unwilling to accept anything less than the T65 cartridge - while they didn't go as far as saying it, the requirements they set out for the cartridge they were looking for effectively made the British cartridges unworkable. Most difficult among these requirements was the demand that the rifles be able to penetrate helmets at extended ranges (something like 500 yards, though I don't have my books on hand to confirm that). While trials accepted the fact that the British cartridge was better for automatic fire from the shoulder, this long range performance was more important when you consider that an automatic rifle - effectively a light-light machinegun - was a key part of the program. Machinegun doctrine generally involves longer-range fire than normal infantry rifles, and at least with the US Army long-range performance for machineguns was historically enough of a concern for the development and adoption of a new model of 30-06 cartridge (M1 Ball).

The British tried to pump more power into their .280 cartridge (by this point the .270 cartridge was dead, as its lighter bullet made that long-range helmet penetrating requirement unworkable), but this created its own problems. As originally designed, the .280 cartridge sat between what are traditionally considered intermediate cartridges and full-powered cartridges like 7.62 NATO. The British added power, but that created its own problems. The much-vaunted controllability in full-auto fire of the .280 British cartridge went away as it became more powerful, and, more importantly, accuracy dropped off unacceptably. By this point, the new .280 cartridge was proving to markedly inferior to the T65.

By this point, the British had nominally adopted their EM-2 rifle in the .280 cartridge, but the deal was far from sealed. The Belgians had reworked the FAL for the T65 cartridge, and between its conventional layout and (arguably) superior ergonomics, it was favored by most non-US participants in the program. In 1952, a quid pro quo deal was made - in return for the US T65 cartridge being accepted as the NATO standard, the US would adopt the FAL - pending acceptable performance in army trials, of course. And here was the problem. The FAL fell short of the T44 in several key areas, namely cold-weather performance, and it was claimed that it would be cheaper for the US Army to adopt, since it on paper was an outgrowth of the M1 Garand and would be able to reuse some of its tooling (a claim that didn't pan out in reality).

Of course, in retrospect, a lot of the assumptions these decisions were built on didn't pan out. The machinegun variants of the Light Rifle contenders didn't pan out - the M15 Squad Automatic Weapon was underwhelming and dropped entirely, and the heavy-barrel FALs (FALO and L2 rifles) were generally considered to be worse than the Bren guns they replaced. Battle rifles were found to be unnecessarily powerful and nearly uncontrollable in full-auto fire, and it wouldn't take long for the US and Britain to once again start looking into the development of small-caliber, high-velocity cartridges.

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The sources I'm drawing from are below. They draw from the three sides of the Light Rifle competition, though admittedly each is biased in favor of their own gun. Hatcher's is perhaps the most authoritative, since he was the former head of the Army Ordnance Department and was most familiar with their methods.

Hatcher's Notebook by Julian S. Hatcher

The FAL Rifle by R. Blake Stevens, Jean Van Rutten

THORNEYCROFT TO SA80: British Bullpup Firearms, 1901–2020 by Jonathan Ferguson