The trouble is employees are already paying 12.4%* of their salary into Social Security, so additional savings would be a significant burden that bottom 90%. If the SS money went into a managed retirement program like* * TSP it would accomplish what you're proposing.
The rationale was stated quite frankly by FDR:
"We put those payroll contributions there so as to give the contributors a legal, moral, and political right to collect their pensions and their unemployment benefits. With those taxes in there, no damn politician can ever scrap my social security program."
He didn't anticipate an aging population, let alone index funds.
Social Security is a wealth transfer from one generation to another, so after a generation has paid out to their elders, they (not unreasonably) feel entitled to collect from it. Even if they were feeling generous, they've generally planned their finances around those benefits. That's why reforming the system has been politically intractable.
* You're taxed 6.2% and the employer is taxed 6.2%. When employers are budgeting for salaries they're going to consider the full 12.4%.
* * TSP is essentially the military's 401k with a group of very low cost funds, only here you would require the assets stay in the program until retirement.