If you’re using a secondary turbine to generate electricity to turn a fan, how is that an improvement from directly turning the fan with your primary turbine? You’re taking a hit on both conversion and weight efficiency in that setup.
Weight wise, it balances out. One helicopter turbine can generate enough electricity for 4 electric motors. The electric motors are going to be much lighter than turbofans, while providing the same thrust, since most of energy from turbofans are wasted as heat. There is definitely more room to improve the efficiency of electric motors. So you might be able to get an ultra light weight and efficient engine setup.