The .223 and 22-250 are fine for deer...The problem through the years has been that the bullets avaliable are for varmints, not deer ...Back in the 70s I killed a few dozen deer with a 22-250 and the 55gr Remington PSP bullets...
Both Federal and Winchester now load bullets in the .22 calibers that are suitable for deer...In the .223 Federal loads the 60gr Nosler Partition and the 55gr Barnes Triple Shock while Winchester offers a 64gr PowerPoint...All of these are suitable for deer...
My niece has killed 9-10 deer with the 60gr Nosler Partition...She gets nice holes on the off side and no deer have run further than 50 yards...