I have shot a number of deer with an AR15. Two of them dropped in their tracks and so much as kick. One was a full on broad side both lungs and heart and the other a head shot. Others have gone maybe 50 to 100 yards and dropped. That said, shot placement and deer size are huge factors. The head shot was a very large mule deer while all the others were Columbian Blacktails. Read your state's hunting regulations to be sure the center fire .22 is legal for big game.
Is the .223/5.56 light for deer? I'd say under most circumstances, yes.
The .223/5.56 puts 'm down because humans do not have the natural survival instinct that deer do. A deer does not know he's been shot with a gun like a human does. So there is a psychological affect that works with humans. There are three ways to take out a human; bleed out, disable the central nervous system and the psychological aspect of oh no, I've been shot! There are only two ways to take out everything else; bleed out, disable the central nervous system.