The 5.56 NATO (223 remington in civvie clothing) is designed to "destroy the enemy". In plain English, that means it was designed to kill big muscular communist humans.
A deer's "boiler" is not significanty larger than a human, especially a human wearing webbing or combat vests, chemical suits etc.
It is just as easy to blow off a leg and wound an animal with a 3oo Win Mag as it is with a 223. The secret to successful humane hunting is to be an accurate shooter and know when a kill is safe and assured. There are those that think that because they have a 338 Lapua magnum means they can hunt at 1000 yards... well, if you can confidently drop a SINGLE round in a 5" circle with a cold bore shot and read conditions, give it a try. Most (including me, a long distance competitive shooter using ultra-precision equipment and optics) cannot. The consequence is having to hike a kilometer away and track a wounded suffering animal. Therefore, smaller calibers with known limitations may actually be better.