One of the sources you provided states the following:
"Users are advised that the student essays were written by 5th graders and should not necessarily be relied on for historical accuracy."
Another source mentions that it, too, was written by school children.
Maybe I'm not as smart as a 5th grader, but let's at least cite sources composed by adults. My sources included letters found in Confederate service records as well as letters to the Confederate Secretary of War. The young men being returned home were in the mid-teens, and I assure you, none were as young as the children you have in mind.
A review of the 1921 census of Confederate veterans in the State of Alabama would be helpful. The census form includes date of birthand a date of enlistment, so age at time of enlistment can be verified on these records.
By 1921 a person born when the Civil War began would have been sixty years old. By this time most older Confederate veterans had passed on to their reward. Nearly all of those who remained among the living were young men when the war began. If children thirteen and younger enlisted in Confederate service, you'd expect to find them here. From personal research with the 1921 census schedule, it's rare to find a veteran born in 1849. I don't recall seeing any soldier's birthdate past that year.
One of your sources mentions U.S. enlistments which include perhaps three hundred service files of boys thirteen and younger. That's 300 out of nearly three million Federal enlistments during the war. If the same ratio of child enlistments held in Confederate service, we might expect 150 to 200 Southern enlistments of boys thirteen or under. I don't know how a fifth-grader might answer, but that very small number seems reasonable.
If I may ask you a question -- let's suppose you had a ten-year old boy who wanted to join the military. Let's also suppose that U.S. military authorities were willing to send him to a combat zone. Would you allow your little boy to leave home to fight in Afghanstan, Iraq or Somalia?