The wedge shape comes only from the space available to pack them in. If there is plenty of space between your magnets, then there is plenty of room for rectangular coils. Of course, if there IS too much room between magnets, then maybe the disk is bigger than it needs to be, and tightening up the space between magnets (and consequently making the coils tighter and then wedge-shaped), you will end up with a more compact, lighter, and cheaper axial-flux generator. Trade-offs.
The "legs" aren't as important as you think. The flux is "linked" when the field passes through the loops of each coil. Once the most flux is passing through the most of the coil area, that is the moment where the flux is a peak. Then the magnets continue moving along until they pass the next magnet pole. Since the field is reversed, the peak flux at this point is the same, but in reverse.
Between those two moments, the flux has gone from a peak at one polarity, to a peak at the other polarity. The difference between the two, divided by the time it took for the magnet to go from one to another, is the definition of the open-circuit voltage the generator will deliver. If you speed it up, the time between magnet passings will be shorter, making the voltage higher. For more detail on the formula, if you're interested, you can look up Maxwell's laws in a physics textbook or on a website like Hyperphysics.
There are some counter-intuitive things about the relationship between flux and voltage, so I won't confuse the issue, unless the details are of interest to you.