Minor technical details were the official justification for the adoption of 7.62 NATO, but the it really came down to a variety of factors.

First there's the question of the new caliber in general. There were huge stocks of 30-06, .303 British, and 8mm Mauser lying around, so why not use that? Well from the outset it wasn't immediately clear what the new cartridge specifications were going to be. FN was doing experiments with 8mm Kurz, for example, while the British were on their way to developing a 6mm cartridge (.280 British) that was considerably weaker (and thus easier to control) than a full-powered cartridge. Additionally, new advances in gunpowder had come along, meaning that similar performance as older cartridges could come from more compact designs.

Though looking back today it's clear things didn't work out as intended, NATO from the outset hoped to standardize equipment - similar to what the Warsaw Pact did - in order to streamline wartime logistics. A huge part of this standardization effort was the NATO Light Rifle competition, which intended to develop a new standard rifle and cartridge for NATO militaries. The design needed a rifle and squad automatic variant among other requirements. From the outset of this competition, keeping old cartridges was clearly out of the question. The rimmed design of .303 British had been recognized as early as 1913, and even the British had been using 8mm Mauser in their semiauto rifle experiments in WW2. 8mm Mauser was a more realistic option, but there were political implications there (using the cartridge of the guys you just beat) and as far as I'm aware it was never seriously considered beyond British wartime experiments. In fact, 8mm Kurz got more traction postwar, with FN designing a carbine that would ultimately become the FN FAL originally around the 8mm Kurz cartridge.

So back to the Light Rifle Program. The Light Rifle was supposed to make use of wartime experience to develop a common modern rifle for NATO forces. A huge part of wartime experience was the realities that combat took place far closer than the maximum effective ranges of the cartridges in use at the time. That led to a lot of inefficiency - extra weight of cartridges, less controllable fire, and harsher recoil on the men. Intermediate cartridges like 8mm Kurz were the result of militaries recognizing this reality. The Light Rifle requirements were a bit beyond the capabilities of existing intermediate cartridges, but the .280 cartridge developed by the British would be designed as an in-between - lighter than a full-powered cartridge, but more powerful than what would today be considered intermediate.

The US, on the other hand, had an ordnance department that had come to the conclusion that any significant performance losses over 30-06 would be unacceptable. They were pushing a Winchester designed cartridge that offered similar performance but was about an inch shorter overall. At the same time, they were pushing for the adoption of a Springfield design for the new rifle and lobbying hard for the adoption of both. Their insistence on performance at extended ranges would force the British to redesign their cartridge to meet these more stringent requirements, but then everything started to fall apart. .280 British suffered issues with accuracy at range and the added power led to the same controllability issues that full-powered cartridges ran into. Meanwhile, FN had adopted their contending rifle (what would become the FN FAL) to fire the new American cartridge. Ultimately, with the US military refusing to budge on requirements that effectively called for 30-06 performance, a compromise was made. NATO would adopt the American cartridge as 7.62x51mm, while the Belgian rifle - the FN FAL - would become the standard rifle. Unfortunately for the Belgians, the Americans walked back on this agreement as well. Trials determined that the soon-to-be M14 performed better in arctic conditions than the FAL, and the (dubious) argument was made that the M14 could be constructed largely on existing M1 Garand machinery (it couldn't, but it would have saved a lot of money if that was the case). So the US never adopted the FAL, and standardization on rifles fell apart. Outside of the US, the FAL wouldn't see universal adoption by NATO as a consequence of FN's reluctance to let West Germany produce them. Instead, the West Germans adopted the G3, which would go on to see use with several other NATO powers. And the French continued to use their 7.5mm cartridge unique to them in the MAS-36 and MAS-49/56.

Fortunately, the jump from 30-06 to 7.62 NATO wasn't a complete waste. The significantly shorter cartridge did vastly simplify the design of actions. Conversions were also more feasible - converting the FAL from .280 to .308 was a much simpler prospect than adapting the much larger 30-06, and where the US failed to adapt the MG42 to 30-06, .308 succeeded in being adaptable to the design in the form of the MG3. The British were also able to convert their Brens to .308 in the form of the L4 machinegun. In addition, it had logistical benefits due to the smaller size (less bulk means more can be carried in the same space).