For those of you questioning whether it's even possible. It is. I can be done from orbit at this point in time and spinning motion necessary is already achieved by the flow of molten iron under the earth's crust that generates the field in the first place. I do acknowledge that it would require more than one loop, but still is more feasible than a manned mission to Mars. Just not as sexy.
NASA did an experiment in space that showed that this idea might work. They strung a wire out from the shuttle and it developed a strong voltage just from passing through the earth's field. But then it burned out.
My fear is that if we actually did build an earth generator as described it would affect the rotation of the earth. You can't get something for nothing. You have to put mechanical energy into a generator to get electrical energy out of it. In this case the mechanical energy comes from the rotation of the earth. The earth is just coasting so extracting energy to make electricity would cause the rotation to slow down. Do we really want that?
NASA was already looking into it? You're right though that there may be unforeseen side effects even Tesla didn't consider, even if the effect is tiny. For those who can't resolve how we start the magnet spinning... The entire planet is basically a big spinning magnet. The sun would fry us if not.