Nope, she's wrong. And what is worse is that she cited a law that does not exist.
Gun clubs are private property and there are no laws that prohibit shooting by people of any age on private property where shooting is conducted, i.e., private gun clubs.
Young persons under the age of 12 cannot hunt independently of adult supervision, which means hands on or within direct and immediate control of said younger person and attendant firearm by supervising adult(s).
Young people, i.e., kids, can shoot at private clubs and private ranges, and on private properties where shooting is permissible, under adult supervision.
I find it necessary to remind a fair number of adults at my club of the specifications and requirements of adult supervision. Some adults think that as long as they are there, that is, at the same range with the kids, it is ok, but it is not. They must be within direct and immediate control of said firearm when it is being shot by the younger person which means they stand directly behind the younger person and are prepared to take immediate control of the firearm.
Nothing says that they cannot shoot at a private club or range unless the club forbids it. She was wrong.