The benefits of this would be a faster exhaust velocity from the fan shaft (since it's not losing energy to the turbine) that would increase thrust, the possibility of using more exotic fuel mixtures, and the ability to use a pulley ratio to increase the torque of the fan and allow for a larger fan blade radius.
It seems to me like having the turbine on the same shaft as the fan increase the losses in energy. Would it still work if the fuel mixture was burned separately powering a turbine that would then spin the fan on a separate shaft connected by a pulley or sprocket system? Then you could either store a separate oxidizer for increased burning efficiency or divert some air from the fan to the turbine for the oxidizer. Not sure yet if the increased weight from the oxidizer storage would diminish the benefits from the additional thrust.
The CO2 and H2O from the separate shaft with the turbine could be combined with the fan exhaust downstream.
To be honest, I know a bit about rocket propulsion but not a whole lot at all about jet propulsion. When I realized that the thrust from the air could be essentially broken down into the part of the oxygen being consumed by the fuel and the part that was composed of nitrogen and unused oxygen, my chem 101 knowledge said that the burned fuel/air mixture, released as CO2 and H2O, would interact very slowly with the N2 and O2 in the other part of the exhaust. Intermolecular forces aren't factored into ideal gases and these are far from ideal gases, but I'm working with what I've got here.
I've tried to research how heat flows from a hot gas, the CO2 and H2O from the fuel-air mixture burning, to a cold gas, the N2 and unused O2 in the same chamber, but I've hit a dead end that my Google research and limited college engineering classes have yet to solve. My chem knowledge would say that the heat transfer between cold Gas A and hot Gas B would be very slow, absent a conductor, but I don't know if that holds up to reality.