That is very hard to believe, unless he shot every single deer right square in the head. Most deer run a little ways with just about every caliber imaginable, some fall, some run, and caliber size usually doesnt negate the outcome.
But back to the question, yes the .22-250 will kill a deer, it'svery popular here in Texas along with the .243, but our deer rarley see 200 lbs. A well placed shot with just about any centerfire will kill deer very affectively, would I choose the .22-250 for northern 300 lb deer ? no, but it can be done if your skilled and limit yourself to take only great shots.
I took a 160 lb 11 pointer this past year with a 15 yd shot to the neck with a .22 hornet, he dropped in his tracks.
Would I have taken that shot on the same deer to the body at a 150 yds ? Nope! You have to be disciplined with small calibers just as bow hunters are with arrows.