The 30-06 is old and dated... but it's not. It's significantly more powerful than the 308 (though I love the 308) and not far from the 300 Win Mag. On perfect shots, anything will work including 5.56/223. It's those weird shots that you don't get to pick that you're glad you brought something more powerful.
I remember a nice buck (aged at 12 years old, it was a cowhorn spike) on an island hunt about 10 years ago. It came running into a grunt call to about 6 yards from me, head down and snorting. I only had a front shot on it and took it with a Marlin 1894 44 Mag loaded with tough 240 grain XTP handloads right in the brisket. I shot through the heart, lungs, liver, stomach, with the bullet stopping right by the tail. About 4 feet of penetration. The deer ran flat out for a full 20 yards and piled up. The "blood trail" was about 3 feet wide and completely saturated the ground like it was laying blood pavement.
A super fast bullet might have fragmented before reaching the heart on that shot.
I've also had bad experiences in my hunting time. I had a phase where I really wanted to hunt with a 5.56 AR. I shot a 6-point buck 3 times (62gr Federal Soft Points) at about 50 yards (it would run a bit then stop) all aiming at the heart one after another in a 5 second period. The buck ran off and there wasn't a speck of blood or hair where I shot. Never found a sign of that deer. I spent hours looking for that deer, even did a grid search for about 10 acres from where I shot and went back the next year to look for a skeleton. Nothing.
I took my daughter out last year for a meat doe. I shot one at 25 yards with a 350 Legend (Herters 180gr soft points). Had 5 minutes to pick a shot, shot high heart from a tree stand. The doe ran off and we got down 15 minutes later to follow the blood trail. I ironically found a perfectly mushroomed bullet laying right where I shot filled with blood and dark hair. Not a drop of blood elsewhere. Spent hours looking for a trail in the direction I saw the doe run. Nothing. Never found a drop of blood or a deer.
All this to say, I've never lost a deer with bigger rounds, but I've lost them with smaller ones on 3 or 4 occasions. Not once have I lost a deer with 30-06, 12ga slugs, 44 Mag (rifle), 348 Win, or 7mm-08.