This can be a touchy subject.
Having said that, yes a 22-250 can kill deer. With good shot placement, and limited fairly close set ranges, this caliber can kill deer humanely. There are some people who hunt out in TX, and a 22-250 is all they use. They are having good results from what they have said. Shot placement is the most crucial element in killing a deer vs knockdown power and bullet size and weight. But having said that, too, remember why use a slingshot when there are more appropriate choices for the job. Personally in my opinion, a 243 is more better suited for taking deer at the ranges you speak of. A 25-06 would be better yet. My son killed his first deer with a 25-06 a few years ago. The 7mm-08, 270, 30-06 or 308 would all work. If I had to pick one out of all of these, the 270 would be hard to beat. Shoots flat, and is perfect for deer sized game.
Brad