Hi Tauno,
The coils would have to be perpendicular to the shaft, not wound around it. For example, you could have several oblong coils wound on flex or baked into a molding and the winding would never loop around the shaft. You could even have only one coil but that is difficult to mount and clean unless you use a flex circuit that can be wrapped around and held at a distance from the shaft's surface. The thing is to create an antenna that has a reasonable effectiveness.
Yes, the air gap can be a problem. But it may be ok if you only need little power and there is no hardcore limit on primary power (not battery powered etc.).
Regards, Joerg