It's not that a .22-250 can't take a deer, it's where you have the shot, and what you may have to shoot through to get to it. A .22-250 is more of a Varmint load than a deer load.
An M-16 uses the .223 and is known to tumble inside a body, which makes it a good load in combat.
A chest shot on a deer with a .22-250 would not be advisable, since the bullet may deflect if it hit a rib or disintegrate all together on a shoulder hit. It wouldn't be out of the question for the bullet to enter the chest and exit from the back end or anywhere, without hitting any vital organs. You really should have a little more weight to produce more penetration and hydro-shock on a bigger animal. Besides, if you have to shoot through any kind of brush or bushes, deflection is another concern with the lighter calibers.