Actually, it is possible to have a sequestrectomy. That's the medical term for taking out the free-floating disc fragments. This isn't usually done because of the concern for a second herniation of the damaged disc.
But doctors are starting to re-think the standard discectomy or removal of the entire disc after herniation. Taking the whole disc leaves a narrow disc space. The ligaments and nearby capsule are affected by the change in disc height. The spine can become unstable without the disc.
A recent study compared discectomy with sequestrectomy. Patients with just the fragments removed were happier with the results. They also had fewer second herniations. More studies are needed to follow patients long-term before a change in treatment will be made.