the EAA had a program called Young Eagles, it may still be running for all I know,
It certainly is still running. I read in various aviation publications about events taking place quite often. The EAA membership continues to support this program in a substantial manner.
Now as to the original question posed as it applies within the USA:
There is no rule or law in the US preventing a passenger from handling or otherwise operating the controls of any aircraft during a private flight. They are not a required crewmember and therefore not exercising any airman privileges and unless they hold a pilot certificate they are not a pilot. It is the responsibility of the pilot in command of the flight to conduct it in a safe manner and entirely within their discretion to determine who shall be allowed to manipulate the flight controls.
Exceptions:
Air carriers operating under parts 121 and 135 are expressly forbidden to allow a passenger to manipulate flight controls.
"Child pilots" on "record setting" flights:
A number of years ago there was a somewhat popular trend involving children too young to hold a student pilot certificate taking credit for record setting flights. They would fly with instructors or other qualified PICs with the intent of claiming records or seeking publicity for their "deeds". Personally I thought at the time that this phenomenon was simply a laughable by-product of well-to-do parents engaging in trendy over-indulgence of their little ones.
Laughable but relatively harmless it seemed to me at the time. Then this little girl and her instructor on one of these "record flights" were killed when they crashed shortly after taking off in the vicinity of a thunderstorm from a high elevation airport in a Cessna Cardinal. Obviously the instructor was PIC, but the national furor over the child's death eventually resulted in a public law being voted in by Congress prohibiting such "child pilot record flights". So while it's still okay to let kids manipulate the controls, it's illegal to let them do it in pursuit of a record. Funny old world...