Remember the 556 was part of a spec set by the military so that the gun and 200 rounds of ammo weighed a certain amount. The 556 was developed to meet that weight requirement and for nothing else.
I don't believe that was why the 5.56mm was developed, and none of the books and Army documents I know of ever mentioned any requirement about the rifle plus 200 rnds weight.
What is discussed is that a faction in the Army, that didn't want the AR15 produced a ballistic requirement (certain velocity at a certain range) that the AR15 could not meet in its original chambering, which was .222 Remington.
That faction hoped that the requirement the .222 AR 15 could not meet would lead to the entire idea of the rifle as the service rifle being dropped.
The .222 Remington Magnum could meet the requirement, but was too long to fit in the AR15.
Another faction in the Army development section got around this by designing the 5.56mm round, short enough to fit into the AR15 but enough larger capacity than the .222 to meet the velocity requirement. And since the AR in 5.56mm could meet the requirement, we got the M16 and the 5.56 as our primary service rife round. There's a lot more to it, of course, but that's it in a nutshell.
Which, of course has nothing to do with the .223 for deer hunting, or the .223 vs. the .44 Magnum.
I wouldn't choose a .223 (with any bullet) for deer hunting, for two reasons. First is, I believe bigger bullets work better, but more importantly, the .223 isn't legal for deer hunting where I've lived and hunted.
For most of the last 60 years, about 2/3 of the states did not allow the .223 for deer hunting. In the last few decades, that number has changed, but there are still about 1/3 of the states where legal deer rifles must be .24 caliber or larger.
There is a world of difference between military use of the 5.56mm and big game sport hunting. Not the least of which is that sport hunting rules are based on what most people are most likely to do, not what trained soldiers with select fire weapons are able to do in combat.
To humanely take game, with any round, you need to be reasonably precise with your shot placement. The smaller the caliber gets the more precise you need to be.