What you are suggesting is actually the direction surgical treatment may be headed. It used to be that the total disc was removed. This procedure is called a discectomy. Discectomy was done to avoid further disc breakdown and a second herniation of the remaining tissue.
But more recent studies suggest that discectomy may actually speed up disc degeneration. Loss of disc height and damage to the end plates (cartilage between the disc and the bone) seem to be linked with discectomy. The end-result is an unfavorable outcome after surgery for some patients.
Artificial disc replacement (ADR) is a fairly new option. Once the disc is removed, a special device can be implanted in its place. This ADR maintains the normal disc height and preserves motion at that spinal segment. If only a small piece of the disc is loose, then a sequestrectomy can be done. In this procedure, the surgeon only removes the disc fragment leaving the rest of the disc alone.
Early studies so far show that sequestrectomy may be superior to discectomy and a good option for some patients.