There's some truth to that, but it's not the whole story. Yea, shoot one in the ear with a 22 and it'll drop, gut shoot one with a 300 and it'll go for a mile.Thats more about shot placement than cartridge choice. You can DRT a deer with a .22lr or one can run 100+ yards shot by a 300 win mag.
Another view of the deer above: He was quartering away, went about 80 yards with his lungs turned to goo and the top 1/2 of his heart looking like hamburger.