A bit more research tonight...
This has (of course) been studied in detail over the years.
https://www.researchgate.net/publication/3462819_Wind_Energy_Conversion_Using_A_Self-Excited_Induction_Generatorhttps://www.academia.edu/20662997/Maximum_power_extraction_from_self_excited_induction_generatorshttps://my.ece.utah.edu/~bodson/pdf/Analysis%20of%20Triggered%20Self-Excitation%20in%20Induction%20Generators%20and%20Experimental%20Validation.pdfI was somewhat intrigued to find a number of papers published in the 1980's by researchers at University of Calgary. Local to me, but probably long gone, and I've never seen a trace of their handiwork up on poles around me here. Maybe a little digging will turn up more interesting local connections.
A wind turbine derived from an induction motor (or asynchronous motor as the academics has put it) has advantages and disadvantages.
Good: It is simple, cheap and robust. It's easily connected to a variety of loads and regulation systems.
Good: it can be operated in variable speed or constant speed modes, depending on the system that loads and controls it.
Good: the electrical circuits can be excited by capacitors.
Meh: Its self-excitation also depends on variable magnetism in the rotor.
Bad: Common 4-pole motors self-excite above 1800 RPM. That's too fast for a wind turbine.
Bad: If the load changes, the upstream effect on the wind turbine is unstable.
Bad: It responds to a resistive load very differently from a reactive load.
The typical solution for the speed mis-match is a gearbox. Those of us who've been watching wind turbine technology for years have seen the WT graveyards in California, many of which suffered from inadequate gearboxes. The ideal DIY solution does not have a gearbox. Matching direct drive rotors that need to run between 200 - 600 RPM to an induction motor suggests that somehow the induction machine needs a 36-pole rotor. I don't know how to do that.
There is a silly idea in my head that maybe the magnetizing current can be stimulated by a step-up transformer on the capacitor bank. Even if there is some sense to the thought, my intuition says it will blow up the capacitors if something isn't done to protect them.
Variable speed isn't a problem. Many DC and rectified-AC wind turbines (like mine) are variable speed, and well-matched to the cubic wind power curve due to this freedom.
Adriaan has already pointed out the solution to pre-excitation: insert a few small magnets on the rotor to guarantee that a magnetizing current in the rest of the rotor is generated by the capacitor bank. They don't have to be large or expendive, though it is a permanent modification to the induction machine.
It feels tantalizingly close doesn't it?