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Two questions:
First, how does the gate provide a positive charge by running a current through it? I really just don't see how this positive charge could be achieved, unless you had some sort of battery in the gate itself and attached a wire to donate some of the electrons flowing through to the cathode of some other battery. Is it possible to get electrons out of the gate by some other means?
The gate of a FET works much like the grid of a radio tube (valve). The gate is insulated from source and drain; the only current flow (if the insulator were perfect) is to charge the intrinsic gate capacitance. Even a MOSFET has some leakage, but it's pico- or femtoamperes. See this discussion on capacitance and leakage.
It is really not charge injection, but charge separation. When a potential is applied to the gate with respect to the SD, the charges in the gate dielectric separate. This separation creates a layer of positive or negative charges just next to the SD channel.
In ideal cases there is no current through the gate.
