That reasoning is appealing but happens to hold some misconceptions.
First, the city of Paris isn’t in charge of the transport system. It is actually the Île-de-France region through the regional transport authority, Île-de-France Mobilités (the local equivalent of London’s TFL, which I happen to work for). Ms. Hidalgo knows it and it is obviously more of a political move, but it shouldn’t overcloud the question: would free public transport help reduce the number of cars on the roads?
The major problem with free public transport is that it creates the perception of a no-cost service. Just as car drivers commonly assume that there is no cost involved in driving their car somewhere. The catch of the car-based system is that the car trip is not in fact free — one should at least include gas, parking space, insurance and depreciation costs, even if environmental cost could also be part of the equation.