Tires on their equipment will be modified so they do not contain air. Most of them will sell in the normal channels, so will get prevailing prices. I live in an area with many Amish sawmills and they all seem to be very similar to me. Not sure how much he would charge if he supplied the log. The Amish in my area use the same techniques and equipment as anyone else.
Any automation is done with hydraulics.
I've seen automatic mills that were all hydraulic. Their rolling stock usually has iron rims instead of tires, which make for a rather bumpy ride. In the woods, they subcontract the logging and trucking. A lot will depend on what their church has to say. A church can be as small as six families.