There is the whole issue of 5.56 is/is not sufficient. You already asked us not to engage in that debate, so I will not belabor it...
There is a certain amount of "wiggle room" or margin for error with a larger caliber. If you are using a .308, the deer will not care if it's whacked with a cheapo 150 Rem PSP, a 165 Nosler Ballistic Tip, or a 180 Swift AFrame. Any decent hit will do the job. (A .308 caliber Nosler Ballistic Tip at 2640 FPS will completely vaporize a catalope sized mass of flesh...... steak tartare)
The 5.56 does not have the luxury of having that sort of margin for error. You need appropriate bullet placement, appropriate shot angle and an appropriate bullet. I have no problem with 5.56 on whitetails, including some of the big bodied North Eastern bucks we get up here. However, I will not consider using a cheapo 55 soft point on a whitetail - These are designed to expand appropriately on woodchucks, foxes and the like. Various hollow points are designed for hunting paper or little critters. FMJ bullets are also not designed for big game. They will take whitetails, but the chance of a bullet failure is significantly higher than I would like to accept.
Some will argue the above points. Yes. FMJ/HP/and 50-55 SP bullets have taken deer. They have worked but they are not appropriate. Think of it this way: Nealry everything can be fixed with duct tape and a big hammer, but there are better tools for many jobs.
There are a few .224 bullets that are designed for larger game. Please use these! They are of a heavier construction, and are less likely to completely fragment. Fragmentation is good in a number of situations, but it is not appropriate for hunting deer. Do you really want lead-flavored venison sausage? The heavier bullets are also driven at somewhat lower velocities, which also translates to more modest, less violent expansion and deeper penetration. These are all good in a white tail bullet.
You options, in no particular order, are the Winchester 64grn powerpoint, The Speer 70 Grn SMP, the Nosler 60 grn Partition, and the Trohpy Bonded Core.
My own particular choice would be the Nosler. They are marginally more expensive than the cheaper Win and Speer. However, they are, by design, a bigger game bullet. Since these will likely be used in faster guns (22-250, Swift, WSSM, etc) they have been constructed to hod together. At .223 velocities you should have no penetration failure issues with the Nosler.
Please note: The little 223 sometimes drops deer dramatically. However, there simply is not the same bullet mass, energy, and massive internal damage that a larger cartridge can inflict so you have far less room for error. A simple double lung shot will collect your deer, but you are likely to have a tracking job before you start cutting venison. Other opt for the neck shot. It's all or nothing: A complete instant success, or miserable complete failure. A buddy took a deer last year that someone had attempted a head shot. It was missing half a lower jaw, and was in rough shape. These are not appropriate hunting strategies.
If you can, opt for a quartering off shot (favorite). Take it behind the shoulder passing through a lung, the top of the heart with the major blood vessels, and the other lung. If the deer does not drop instantly it will drop within sight. Quartering on works just as well.
Frozenny