It's probably not a legitimate concern between two advanced regular militaries who follow the laws of war and would bury your dead with respect if they can't actually return them to you, but if you left a uniformed western body where ISIS can get it then I'm sure that would end very nastily indeed, with photos on social media and so on. Think of the military contractors burned and hanged on that bridge in Iraq.

'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.