You may need to correct your thinking somewhat. There are no such things as doubly fed induction generators. What you have there is a doubly fed synchronous generator.
You are right saying that you can compensate for over/underspeed by supplying the slip frequency to the rotor and that is a very common design that makes operation at varying speed possible. It has a huge drawback, though. The PWM fed to the rotor leaks from rotor windings to rotor iron and causes problems with bearing currents. It also causes arcing between brushes and slip-rings. The latter is caused by large current peaks (the winding capacitance charge/discharge currents) and, since ordinary current clamps do not show those peaks, the brushes are designed to handle the design current - not the peak currents. That leads to premature brush failure and quite often to eroded slip-rings.
Now the leading power factor: As long as excitation produces a voltage that just can deliver the needed power, the power factor is 1, which is the same as phase angle zero.
If you increase excitation, the generator's voltage increases. The generator will then deliver its peak current (crest of the sine) earlier than before - simply because the voltage is at its peak earlier. Leading current is capacitive power factor and there you are, generating reactive power.
If you reduce excitation, the peak comes later and the generator consumes reactive power. Which isn't very useful.
I made this as simple as I could. I could have added the beer/froth analogy or used complex numbers. I didn't. Others may do so.
Gunnar Englund
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Half full - Half empty? I don't mind. It's what in it that counts.