The .308 works well in semi-auto short action rifles. That is the primary reason it replaced the .30-06 which it was similar to.
It was adopted by the military as essentially the .30-06 replacement, the 7.62x51mm. The .30-06 was considered the ideal battle rifle cartridge proven throughout World War 2 and prior, but was a little too long for an ideal acceptable weight and reliably full-auto rifle utilizing a box magazine.
It was a matter of reliable cycling in an automatic at high rates of fire.
The .30-06 is still more powerful in some loadings because of the additional case capacity, but it was close enough.
But the 30-06 came from a day when shots at long range were taken by most troops in battle. That would become less common in modern warfare, and long range shots became a specialty filled by some rather than the typical shot for all infantrymen.
This meant such long range ability was unnecessary as standard.
With the adoption of select fire weapons and high round count even in semi-auto, limiting the amount of rounds that could be carried with larger long distances rounds seemed less ideal.
So even though the .308 was used as the replacement for the .30-06, the role of such a round was no longer needed by most infantryman.
(The adoption of body armor by most military forces may reverse this though. With additional power being necessary not for the same long range shots of the past, but to be effective against standard body armor of modern forces.)
Then the rifle it was intended to be used by, primarily the m14 (and m-60 light machinegun) had a very short span as the primary armament.
The US (and NATO) switching over to the 5.56x45 round in the m16.
As a result the military had plenty of relatively new platforms in .308, or more specifically 7.62x51 left over. Not just the USA, but other NATO nations who had the FAL and the G3.
From which most marksman rifles were produced.
Yet the popularity of the round by the military also led to its popularity with civilians. For several reasons, including the same ones that cause people to gravitate towards the AR-15 and .223/5.56x45 today.
Many fine civilian rifles became chambered in it.
That popularity in civilian circles led to it even being used in many quality bolt action designs (ironic considering it was primarily adopted for cycling in semi-auto firearms). That use combined with already being a standardized cartridge resulted in it going full circle. Resulting in even such bolt-action designs being adopted by the military in the caliber. Like the Remington 700 being adopted as a sniper rifle.
So no, it is not so much for its pure ballistics, if that was the case they would have been better never deviating from .30-06. Instead it was just a natural progression of ideas going back and forth between military and civilian. And the round already being a standardized NATO round. Which they try to keep limited for logistical purposes.