You can just chalk it up to culture, but there is also a logic to it that is every bit as logical as the "leave my meat where it falls" crowd: if you have a culture of retrieving the bodies of the fallen, no matter the difficulty, then that means that you won't leave behind the wounded to save your own skin. This, in turn, makes it easier to trust and to risk your life for your comrades -- they'll take care of you in turn, living or dead.
I always assumed the "no man left behind" spoke more about the excessive abilities of the group. (We are so powerful that we can even overcome our enemy to retrieve our fallen men.)
In fact, when for instance Taliban or Daesh look at things, the enemy bodies are not plentiful at all. I think it is extremely rare for an American, British or NATO partner soldier's body to fall in their hands. The Western losses in those wars are very small by historical standards, almost negligible, and evacuation is extremely efficient.
For instance, take the current Afghanistan war. Wikipedia: "As of October 1, 2015, there have been 2,326 U.S. military deaths in the War in Afghanistan. 1,856 of these deaths have been the result of hostile action."
1,856 casualties from enemy action in about 15 years. That is about the same number of casualties as Omaha Beach took in a single day. It's half of what my father's brigade inflicted in a single day against one single Russian infantry regiment in February 1942. It is one thousandth of what German Feldheer suffered in all of WWII, and one five-thousandth of what the Soviet army suffered in their "great patriotic war" during four years.
Casualty numbers in modern wars where Western troops attend are very small, when compared to historical wars.
The current Syrian civil war is then another matter. Its casualty numbers are not big when compared to WWII, but they are much bigger than Afghan or Iraq wars.
It's basically "we are all in the same boat", which is true (sometimes literally) when you're engaged in high-risk activities. Note that high-risk in this case doesn't mean that you might get killed, but that there is a low margin for error leading to catastrophic failure. Special operations are not about being a fair fight, it's about using your advantage and once you loose that you're often in big trouble.
Say you're doing recon on enemy territory and calling in airstrikes. You're a small mobile force that is hard to find having disproportionate impact on the battle, which is good. Then someone drop a glove or breaks his leg or anything else that compromises the mission and now you're just a small force soon to be a disproportionate loss, which is bad.
'No man left behind' is practical, both for the propaganda reason above and unit morale, not about showing off. But like most things in military life I never heard anyone ever bombast about 'no man left behind' like they do in the movies.
Finland was unusual in the context of WWII that all bodies of dead soldiers were retrieved if at all possible, and sent back home, not just buried at a local military graveyard. This diving thing (Finnish divers in a Norwegian cave) has echoes of the same.