Why weren't the rotor tips of the Djinn directly fed from the turbine's exhaust?
The soot in the burned exhaust will eventually clog the ducting. The higher temperature will deform the rotor, or tougher heavier materials will be needed.

The stream of a  high BPR turbofan (ignoring the size) when channeled into narrow ducts will raise the stream velocity, but lower its pressure, which will be an overkill in terms of thrust needed.

Therefore "slightly warm" clean air is best.