SmallTownGuy
Since the EMF is radial to the spinning disk, replacing the disk with a coil shouldn't have any effect at all. The increase in path length should just be cancelled out by the spiral's winding angle.
You're welcome to try it, though. Maybe there's some other benefit (cost?) nobody's thought of.
You may be right given electricity is a right angle phenomenon. I was thinking about other benefits as well and I think that with a coil of heavy gauge wire, one could remove the peripheral brush and place it on the axle where less surface area passes per rotation. Or when you stack them back to back so the current reverses in one coil, the peripheral brush(es) could be eliminated all together.
Not to mention copper wire/cable is cheaper than a solid 1/4" copper plate at 6" diameter.