Contrary to popular myth, 223 rounds can put em down just fine. Being a deer is about the body mass of a man, and the US military has deployed 223/556 rounds as anti personal rounds for decades, the simple truth is they work just fine, and better being deer dont tend to wear body armor. The average 223 round leaves the muzzle between 2500 fps and 3200 fps. With the lighter hollow points putting 1/2" holes through 3/8 steel plates at 50 yrd shots to the heavier Barnes 72 grain solid copper rounds that punch holes in the same plates even at 100 yrds, these rounds are plenty deadly.
Most men want to be more manly, so there in lyes the myth of bigger is better. Not always so. Soft wet bone is not the same as steel, and can absorb rounds faster. Bone material pushed though the wound channels also aide in destructive forces within vitals.
Exit wounds from larger caliber rounds do help with bleed out. The rapid loss of blood pressure causes the animal to faint if you will before dieing. No exit wounds from smaller calibers allow the bleedout to happen internally. Pressure internally on vital organs from the bleeding causes an animal to pass out just as fast.
Shot placement for a smaller round is important. Keep them dead on heart or liver. Also from a more tactical point of view, a shot to the backside base of the skull will put them down instantly. However not a great placement if your wanting to keep the head or antlers intact.
One major benefit of a small caliber round is the lighter recoil of the rifle as well. Lighter recoil will play a vital role in the phycology of the shooter. The shooter will not flinch as much if at all knowing the shot wont hurt them. Big guns make big booms and hit the shooter just as hard. Its actually harder to "Be prepaired" for a recoil than it is to aim and pull the trigger.
Too many hunters miss from the flinch and not from the gun being off its mark.