Will do!

Alright, so I followed the same tack for a while, but the tricky thing is that a hyperboloid of one sheet is doubly ruled, which means you have to worry about secant lines corresponding in some way spuriously, just due to relation with the ruling lines. That inspired me to go in a different direction.

To make a cone, you need the wiper/striker to be coplaner with the axis or rotation; otherwise, it will become a ruling line. This is both necessary and sufficient, given that we don't particularly care about the exact cone. Furthermore, any two non-coincident intersecting lines make a plane, and gravity offers an easy way to produce parallel lines.

You make a half-mould with a cavity that will look something like a speech bubble. The inner edge of the tail(?) will become half of the wiper, and the rest will become half of the upper beam. By reusing the mold, you can ensure the mated halves will be fairly symmetrical bilaterally, which means the center of gravity will be close to coplaner with the meeting point where the wiper forms. Some combination of filler or adhesive in between and a draw string around the outside seems like the best way to hold them together. True, that may introduce asymmetry, but with such limited tools compromises have to be made. The string or rope itself is relatively light, and with attention to detail it will still end up reasonably close to balanced.

Now you have a rigid piece with a center of gravity (roughly) coplanar to the wiper. You can use the parting line to center awl points, which will also be (roughly) coplanar. Tho only configuration of the two rotating awl points involved and the rigid body's center of gravity which can balance is the three of them all lined up along the zenith. So, if you can balance these two components, you've guaranteed that the axis of rotation is the zenith, the line from center of gravity to upper awl point is the zenith. Then, the zenith is in the plane of the wiper, and so the axis or rotation is coplanar to the wiper. QED, to whatever degree that applies to caveman work.

The actual process would be very involved. The task is picky, the pieces are heavy and the top beam is probably made out of clay, which is physically delicate. The way to go is probably to sit them all in a frame with vegetation for padding, move the frame slightly away somehow, and then adjust the frame corresponding to which direction the assembly falls into the padding. It's probably worth it to use knapped clay points for the rotating stake just to reduce sticktion. Experimentation is needed, but unfortunately I don't really have a good place to do it.

Once you're done, easing the beam upwards and then fixing it tightly in place will allow you so start turning. There would still be some error, of course, both from the sources I mentioned and others, but that seems unavoidable. As long as the finished product is fairly close to a cone you can rotate the resulting bearings together dry in order to lap them to a tighter fit.

In case you're curious about the next steps, you need cones in the first place because clay shrinks when drying and firing. Two similar cones will continue to fit together after rescaling, but hyperboloids won't. The plan then is to use the turned cone as a master for female cones, which will themselves be used to make matching male cones. The two bearings will be placed large ends together, and held on a thickened section of the wooden axle, and any stationary frame, by rope tension. The male cones will then protrude out due to being smaller, allowing clearance for the axle to continue on.