While the 22-250 is a great flat shooting round,I prefer the 223 over it. Unless you get a custom barrel for a 22-250 with a faster twist rate,you are limited to shooting lighter weight bullets out of it,60 grain and lighter.
Depending on the brand/model of 223,you can get 1-8/1-9 twist rate barrels,and then be able to shoot heavier bullets out of it,thus giving the shooter more options to choose from. Plus 223 ammo is a lot cheaper than 22-250.
The 243 is an excellent caliber also,but it is also more expensive to shoot.