There is NO set age where any human can handle a .30-06. I've known a bunch of adult he-men who can't shoot it well.

If you have any doubts about the .243, load some good 120-130 grain bullets in the 6.5x55, whether Nosler Partition, Barnes Tipped TSX, or whatever. 2700-2800 fps will do fine, and not kick much more than the .243.

There's no need to use heavy bullets in the 6.5x55 if you're using premium bullets. All they heavy bullets will do is kick more.

“Montana seems to me to be what a small boy would think Texas is like from hearing Texans.”
John Steinbeck