Old geometry needs a different setup and flexibility versus newer. Below is Merckx. This was the best pic I could find with him on the hoods. Note how far back he sits and how much lower the brake hood position is versus late model bikes.



I agree that your son has to rotate his pelvis but he may lack the flexibility to do so. More than likely the reason his back is sore is that a rounded back position reduces / eliminates the glutes and makes power production quad centric. Once the glutes are eliminated, the back muscles work a lot harder to counter the forces from the quads. And the back muscles are designed to keep the back stable and erect versus countering hundreds of pedals strokes.

I think the fix is a smaller old school bike or raise the seat and move it forward and put in a smaller stem and raise the bars. Old school stuff as I remember it does not have a lot of flexibility to change effective top tube length.

Have your son try to visualize a very light touch on the handlebar and to squeeze his shoulder blades together. In other words, try to raise his chest. This will cause his hips to rotate - hopefully.