There would obviously have to be a transitionary period where the government makes up the shortfall, in order to honor the commitments made to retirees. It would be painful, but only a small addition to the 87 Trillion in current liabilities of the U.S. government.
http://online.wsj.com/news/articles/SB1000142412788732335320...
Thus if you replace these systems with mandatory investments, you are reducing their available cash.
Everyone who gets a paycheck pays their 6% in social security, regardless of income level.
Social security tax is 12.4% on the first $113,700 of wages or self-employment income, and 0% on income from all other sources and wages and self-employment income past the first $113,700, so, no, you don't pay "6% in social security, regardless of income level".
(Note that on wages, but not self-employment income for obvious reasons, half of the 12.4% is paid as 'employer share', so that the employee never sees it, so it might appear, if you look only at wage withholding, that the rate is 6% -- or rather, 6.2% -- but that's not accurate.)