The old stitch marks may show when letting out the seams due to the cotton content. You could open a seam and test to see the result and decide if you can live with it. The best way to deal with the old stitch marks is to open the seam, apply water to the marks. scratch the stitch marks with your finger nail and press from the inside (ringside of the cloth), not on the face of the cloth. Do not let out the center back, release the underarm on the side body.
This tailor had originally recommended letting out the CB before they realized it was partially cotton.
Was this bad advice? Should I be taking this to someone else?
Will releasing the underarm fix the tight upper back though?
Probably but you haven't posted a picture and my response is a general one rather than a specific. Generally letting out the center back only results in creating a mess. Letting out both sides at the side body makes the armhole wider and releases the back enough to create more motion.