Putting iron cores in the coils reduces the effective gap, allowing considerable additional flux for a particular set of magnets - up to the point where the iron saturates, beyond which any additional flux "sees" the gap as a pretty good vacuum.
But there are a couple problems with that:
First: An air core has no "cogging", while an iron core will have a lot of it unless you get the geometry just right. This is because the magnets attract the cores and prefer to be over them to being over the gap between them. This isn't an issue (except for extra vibration) once the mill is spinning. But starting it is like pushing a car down a level road that is all bumps - you have to push HARD to get it over the first bump. That means you need some minimum wind to get the mill to spin. But while lift-type blades are very efficient and have a lot to torque when spinning they have very little when stopped - just when you need it most to overcome cogging and start the mill after a calm.
For battery-charging applications you can accept some cogging. There's a minimum voltage you must exceed before the battery is charging, which means a minimum speed ("cutin speed") for the mill and a minimum windspeed to spin it that fast. If you've got your cogging down to where the mill starts at a windspeed below that necessary for cutin, you're OK. For heating there is no minimum: If you can spin the mill you want the power, even if it's a small amount. So you'd like the mill to start at a very low windspeed.
Keeping cogging low a radial-flux machine means careful design of the shape and position of the cores, and superb execution when putting them in place. It's hard to calculate and hard to do. Without cores you just have the coils. Coil shape and location affects the output, but it's NOT critical - they'll work just fine when significantly out of shape and position. And without cores the only thing keeping the mill from spinning is bearing friction, so it will start at a breath.
Second: Cores have losses: Iron losses - from the energy lost as heat when flipping the electron spins - and "copper" losses - from the energy lost to resistive heating by circulating "eddy" currents induced in the metal of the cores. To minimize those you have to chose an alloy with low residual magnetism and build it in layers ("laminates") with insulation between them, oriented correctly with respect to the magnetic field motion. (Look at transformers and generators to see how it's done.) More critical alignment, more complex design and construction work.
The power lost to the cores comes from the mechanical power that you collected. It's that much less that you have available at the output. More importantly, the losses are dumped as heat into the stator. The limit on an alternator is its ability to lose heat, to avoid melting metal, degrading insulation, and warping structures. And the main downside to the homemade radial-flux machines is the difficulty of dissipating heat. The heat from the cores is heat you can't dissipate from your coils. If you leave the core out the only heat dumped in the stator is the copper losses from the real copper wriing - the unaviodable resistive heating from the output current and a trace of eddy-current heating.
Since all you are using the core for is increasing the magnetic flux through the coils, you can achieve the same effect by leaving out the cores and using big magnets - making them thick enough that they can drive the flux across a significant coreless gap. Which is how the people who populate this board tend to design their axial-flux machines.
(Radial-flux motor-conversions don't have such an issue with heat: There's plenty of surface area to dump it, lots of airflow to dump it into, and the coils aren't embedded in plastic. The motor core geometries and alloys have been designed and debugged by engineers over more than a century and are mass-produced. So converting is mainly a matter of building a replacement rotor with permanent magnets that doesn't cog badly - in a surrounding geometry where that's relatively easy to compute - and perhaps rewinding the stator according to well-known rules. You can accept the core losses because they're not that great, you get some of it back by being able to run the coils cooler, and can compensate for the rest by making your blades a bit longer. So you end up with a machine that can handle a lot of power gracefully, without burning up.)
