I've had similar situations happen--with me! Sometimes, when you gotta go, you gotta go! That's especially a problem when switching over from home food to the backpacking diet, which usually contains a lot of dried fruit and nuts. In those cases I "go" first, then dig a hole alongside, push the waste into the hole with a stick or stone, put the loose dirt back on top and press down, and carry on. It's important to teach the kid to do this himself and to pack out his TP. For quite a while you have to check afterwards to make sure everything is completely buried.

Having seen obvious human waste--unburied and with free-floating toilet paper--right next to a trail (aargh) any number of times, I wouldn't worry. At least you made sure your son's was properly buried!

Not everyone has full control over bowels and/or bladder at the age we think they should. I was a late bed-wetter and so were three of my four children (#4 basically toilet-trained herself and was dry at night by age 2, making up for all the others) and all six of my grandchildren, two of whom were 12-13 before they gained complete control. No medical problems and no negligence on the part of the kid, just what the docs called an immature bladder. The same can be true of bowels. IMHO, parental pressure only causes more guilt and shame than a kid with such problems already has--believe me, they are already well aware that they are not "normal." I therefore am in full sympathy with your son! You can gently and unobtrusively keep track of when he "goes" and gently suggest he try at the same time the next day, but please don't press him too hard!

Far more important than privacy or being far from the trail is being well away from any water, including seasonal (dry) watercourses. I'd aim for that goal rather than privacy.