I am not sure what your point is. Private accounts do nothing to reduce the actuarial deficit in the system.
They do because you still pay into the system but draw less from the system because you are withdrawing from your own savings!
consider the nuance:
returns only slighty better than the ultraconservative figures assumed for discussion purposes today would greatly ease the consequences of progressive price indexing or any other means of addressing the actuarial deficit
if the returns are worse, we'll be fully socialist before any of this even matters and after much blood in the streets