Worn tie rod end could allow enough back and forth play to develop a shimmy/vibration. Felt in the steering wheel, and could also cause a knock/clunk noise over bumps.

Bushings are generally pried on for excessive deflection and visually inspected for cracks. Somebody could mount a little video camera in there to monitor for play in various components if it is that hard to figure out. When I worked at Ford I saw some engineering videos from under-car cameras, and it was very surprising how much force was being applied to certain things during regular driving, and how much they moved under load. It is difficult to duplicate real driving loads in the shop.

At the least, I suggest having a non-tire shop indy diagnose it. Sounds like the shop that you just mentioned might be worth trying, as they have shown a lack of greed by refusing to work on something they don't think is the cause. Maybe they are willing to take a fresh look at it.

Certain tire faults can go undiagnosed, such as excessive road force variation. The tire may balance out fine and be round, but can still cause vibration. Usually excess RFV is more common on lower end tires, but it is always a possibility. Not all shops have a machine capable of testing for it. For more info on this tire issue, check out gsp9700.com. You can see if your local shops use this equipment. Often they will only use it when necessary or they may charge extra for it.