Q: Should my son be putting spin on the ball when shooting?
A: Yes, I believe spin is important to stabilize the flight of the ball aerodynamically. But "trying" to get spin is not the answer.
First, have your son observe how he shoots (without trying to put more spin on the ball). Is he flipping his wrist or somehow using wrist, hand or fingers to power or guide the shot? He probably is, and that's why there's little or no spin. That action interferes with any natural backspin that's possible.
My way of shooting is to power the shot with the whole body and make the Release a simple, upward "straightening of the arm" rather than any wrist or hand action. If the wrist and hand are relaxed when you straighten the arm, the hand will bounce. That bouncing is one of the hallmarks of great shooters.
Work with him and see if he can develop a Release that is just that ... a straightening of the arm with relaxed wrist and hand. The direction is very high, high above and directly in line with the basket. Start in close where, with no legs, he can swish the ball effortlessly with this motion. At age 13, he should have enough strength to shoot a high arching swish shot with just the Release muscles from 3-4 feet. His Set Point is best if in line with the shooting eye and above the eye, if he's strong enough. If he's not, then have the ball below the eyes, but still in line with the strong eye.
Make sure he's shooting "up" in order to come "down" softly into the basket. This upward, pushing action is done at about 70-75% of max. speed and force, a motion that will not hurt his arm but provides a quick, high-angled Release.
When he just pushes the ball with the arm like that, with a relaxed wrist, he should get perfect, medium backspin. (The ball is held with a little bit of pressure in the finger pads, which helps in the Release to give the backspin as the ball rolls off the fingers.) Once that's learned, then as he moves further away from the basket he needs to add more and more body/leg action to power the shot and allow high arching, effortless shooting, keeping the Release speed and power always the same.
My video shows exactly how to do this. This instruction should get you started. As you'll see, this is a different way of powering his shots and will result in more consistent, high arching shooting.