Part of the problem in old people is that they have too much memory in the immune system, especially of pervasive herpesviruses like cytomegalovirus. Those memory cells take up immunological space that should for preference be occupied by aggressive cells capable of action.
Another point: in old people, as a treatment for immunosenescence, immune destruction would probably need to be paired with some form of cell therapy to repopulate the immune system. In young people, not needed, but in the old there is a reduced rate of cell creation - loss of stem cell function, thymic involution, etc. That again, isn't a big challenge at this time, and is something that can already be done.
At present sweeping immune destruction is only used for people with fatal autoimmunities like multiple sclerosis because the clearance via chemotherapy isn't something you'd do if you had any better options - it's pretty unpleasant, and produces lasting harm to some degree. Those people who are now five or more years into sustained remission of the disease have functional immunity and are definitely much better off for the procedure, even with its present downsides, given where they were before. If the condition is rhematoid arthritis, however, it becomes much less of an obvious cost-benefit equation, which is why there needs to be a safe, side-effect free method of cell destruction.