The problem with the small-caliber, light-bullet cartridges as deer-cartridges is that of a bad hit. In the neck, or in the heart/lungs, smaller deer drop quite readily. A hit on a heavy bone can result in a surface wound, with the deer escaping to die unfound, wasted. Same for a gut shot...
I'm not saying this can't happen with a .257 or an '06, but it is less likely.
Of the three, the .243 is the best choice. With the 100-grain bullets, the .243 has been proven to quite effective on white tails. I've taken some 20 or so, myownself.
Mule deer? I go to my '06.
Regards, Art