Wool wheels, a/k/a 'walking wheels' and 'great wheels' were often made by wheelwrights.
Flax wheels generally had much smaller wheels, and came in a fascinating variety of forms, one called a 'chair wheel' because it appeared to be mounted on a chair-like base similar to a Windsor chair.These could be made by turners or joiners.
So the occupation "spinning wheel maker" need not be what appears in a US Federal Census enumeration.
It is too bad a photo or detailed description of the wheel are not available.If someone wants to send me a well-lit photo, I could post it on my coverlet-related blog for all to see.
frostfree12 at yahoo
Meanwhile, here is a site in Bradford Co, PA (where there were, ahem, also Schoonovers) about textile tools, including spinning wheels: