According to Dr. Mike English, NMSU Extension Entomologist, once the roots are infested, the tree is doomed to a slow death. It is not possible with reasonable means to eradicate the aphids from the roots. In time the tree will die. Initially individual branches will die and leaves will drop from other branches. In spite of infestation, the tree can continue to grow and produce apples for years.
Crown gall is a disease of apples and other members of the rose family.
This disease causes the formation of galls on the roots, base of the trunk, and sometimes on the branches of the trees. It may be confused with the wooly apple aphid galls, but the white, waxy covered, insects are not present. Since you described the presence of the insects, they are the most likely cause of your problem. Regarding your plan to dig the sprouts to start new trees, these sprouts are from the rootstock and will not produce the tasty apples you described unless the tree was started from seed and never grafted. If these are from the rootstock, the new plants would need to be grafted with the variety that you like.