243 is more accepted among deer hunters than 22-250.Hi. Check your local hunting regulations - in the states where I have lived, bullet diameter must be greater than .22 caliber in order to legally hunt deer.
I think that the 22-250 is a fantastic round for varmints - shoots fast and flat. The .243 would make a better all purpose rifle for your proposed uses. In addition, there is a large selection of bullets (type/shape/weight) available n .243, especially if you decide to hand-load.
Blessings
Paul
More deer have probably been killed with .22 long rifles than any other caliber but that doesn't make a 22 rimfire a humane or effective deer cartridge.