Help needed building an onager

[P. Clodius Secundus]

Wouldn't making it thinner in the middle make the drum weaker, with the maximum amount of force being excerted on the thinnest portion of the drum?

Yes, that is a valid concern with this design. Provided that the center portion of the drum was no thinner than the axle ends that fit into the pockets in the side beams, it would be essentially the same strength as the winch in Duncan's Osprey illustration noted earlier. One major difference is that I would start from one large log shaped into an hexagonal timber and then lathe turn the axle stubs and center section, leaving a short hexagonal section on each end where the holes for the winch poles could be inserted. This further simplifies the construction and set-up of the machine by eliminating the problem of how to attach the drum ends to the rope drum. This one-piece winch would simply be clamped into place between the side beams when the frame is assembled and held there with the clamping force of the stretched spring bundle. When it's time to disassemble the machine, all you have to do is remove the spring cord and knock the side beams apart with a mallet.
For your educational and safety needs you could easily sandwich a ratchet cog between the side beam and one of the hexagonal sections.