I'm of the idea that all 3 calibers are good for deer at the prescribed distances you have mentioned, but personally I am more inclined to choose the .243 caliber based on my opinions and observations over 50 years of hunting only. The other two are pistol calibers that may or may not maintain enough velocity to do a pass through penetration on deer at 125 yards, at 50 yards yes but not 125. Of course bullet selection is key where the base of the bullet is most important because sectional density is what will give us the penetration required along with velocity. One can expect a gain of 500 fps when shooting from a rifle over a handgun.
Another factor is I have witnessed deer running a couple of hundred yards with a heart shot and if the bullet did not pass through it makes tracking very difficult as there is not much of a blood trail to follow since all the bleeding is done internally.
With a .243 caliber you now have a rifle cartridge, that can be loaded with the heaviest bullet offered for that rifle with minimum loads that will reduce recoil significantly for a younger boy but will have better sectional density with velocities of over 2000 fps providing enough power for the bullet to go through in case a heart shot is made only missing the lungs, where now you can have a blood trail you can follow. Most important is that it is a rifle that they can grow old with and as they gain wisdom and experience, they can go on hunts that may require longer shots due to differences in terrain or can be used for antelope and pronghorn and such. Just my 2¢.
