Its been said, but its important. The 22-250 will kill a deer. Now, there are assumptions: good shot placement, the right bullet. If you cannot ensure #1 like hogheads gal (who I admire for her shot discipline!) then you have absolutely no business taking this caliber after deer. Period.
For the second, bullet selection is very important. The archer mentioned above that with 70 pounds of pull on his bow, he can take deer. BUT, his broadhead is opening up 1.25" of wound channel for that soft tissue damage. The bullet has 2 ways of making damage: direct tissue damage and shock damage. To understand this, lets mention some of the other inputs above.
If the bullet completely travels through the deer, it is NOT transferring ALL of its energy. Why? Because it is still using kinetic energy to continue travelling beyond the animal and transferred its remain energy into whatever it hit on the other side. (I think its Newton's law of energy) When the bullet leaves the barrel, it has a specific energy based on mass and velocity. As it travels, air friction reduces the velocity of the bullet, reducing energy. You have to account for all energy in the equation. When it hits the animal, it takes energy to go through the skin, break bone, mushroom the bullet, etc. The perfect transfer of total remaining energy in the bullet at impact would be a perfect mushroom that rests under the skin on the opposite side.
This is the reason why people shooting high velocity bullets without expansion characteristics matching their game find themselves tracking animals all over tarnation. I had a gent in FL hunting deer with a .338. He shot 3 deer over a weekend, never found one. Another shooting 7mm Rem mag in WA. Same results. Why? The shots seemed descent enough. Because the bullets they were using were intended for bigger game. They didn't get the expansion and resulting energy transfer into the animal and really did nothing more than shoot an FMJ through the animal (even though it was a "SP). This is what happened to the 7mm Mag scenario above. With todays bullet selection if you handload (which I don't but read about), using a large, high velocity caliber is doable for smaller game if you use something like a ballistic tip or rapidly expanding bullet.
Now, back to the 22-250. If the shot is marginal, will you get the collatoral tissue damage as you would with a .308, 7mm-08, etc with similarly matched bullets that can do enough damage to bring the animal down? No. That's why bigger can be better to a point.