Feathering the blades is windmill "gearing". It matches the power available from the wind to the power that can be absorbed by the generator. In the overpowered case, it scales back the power converted from the wind: it "spills" some of the wind energy by making the blades less efficient so that less power is generated (totally feathered results in no power, but the blades can be feathered to any degree necessary).
Adjustable gearing would add a lot of complexity, weight, and reduce reliability. Reducing efficiency by (partially) feathering the blades is a much better method, especially since it is already a necessary capability of the windmill.
"Switching gears" shouldn't make a difference unless it brings the blades to a halt. There's still (roughly) the same amount of energy, it's just producing different voltage / amperage based on your gearing. Rotating the blades slightly, however, would control power generation without trouble (though possibly more cost / repair / etc, unless they do this already?).
Good question, but the danger isn't in overloading - they can shut off the power they're getting easily enough. The problem is it's a tremendous waste.
Wind power is good, it's just extremely unreliable and storing it is a harder problem than you'd think. One of the regular users here, DaniFong, posted some good comments and articles on how difficult it is to store power. If you started at her profile and clicked around, she's made some quite insightful points on the topic:
Why can't someone just build a wind turbine that has the similar function of switching gears like the mountain bikes do?
If the blade moves too fast it can switch gears and makes it harder to spin the blade - wouldn't that produce less energy and fix their problem?