I am all for setting good quality standards in performing public functions. The problem is that you are selling me a fake argument. 
The numbers that you cite from a “reputable” organization is chaired by NSTAR, a Boston-based utility company with a direct conflict of interest in municipal outsourcing. See the list of conflicted entities funding Proposition C and the mayor.
You are presenting the Review Board as if it is the epitome of public oversight and accountability. So, who selects the departments to be reviewed, writes the RFP, sets the criteria, selects the finalists, and finally approves the contract? Who monitors the contract, evaluates performance, and penalizes contractors who waste taxpayer dollars? Who has the authority to revoke a contract in-house if the initial assessment was faulty? The Review Board does NONE of these things. It is merely a pawn in a predetermined political game.
You are claiming that services performed by “classified” employees are not outsourced. That’s false: functions performed by classified employees have already been contracted out in Wastewater, Real Estate Assets, street repair, etc. In the budget, the mayor has left these positions vacant and simply out-sources the work.