In the final analysis it was simple really. The back of my desk chair has been loose for years and I thought I had fixed it by shoving a doorstop in between the chair bottom and the metal slat of the back that slots into the groove underneath. But, it didn’t work, maybe the doorstop was not big enough. Anyway it was still wobbling and difficult to sit on. I knew the reason was that the thread of the screw on the metal part under the chair itself was stripped and so it would not hold the back in place. Then, as usual while sleeping, I had an idea, it was there when I woke up, a simple idea. Why not use a bolt and nut in place of the large screw, as long as the bolt was smaller than the diameter of the screw hole it could be tightened and hold the two pieces, chair bottom and back, together.
So I found two medium sized bolts with nuts and turned the chair over. But wait a minute, I saw that it’s not that easy, how to get the bolts in place and be able to tighten them. I realized that to do this I would have to separate the chair bottom from the metal plate that supports it. To do this I would have to remove four large screws that attach the metal plate to the chair bottom. Two screws came out easily, but the third was difficult and the fourth impossible. I used oil (SW40 spray) and some hammering and multiple screwdrivers, until finally it gave. Then to the metal bottom (with the legs supporting it) I attached the metal slot of the chair back onto it using the two nuts and bolts and then tightened them. Then of course I realized that I had pushed the back too far in, so had to release the bolts and push the back back, and then it fitted into place and I could screw the chair bottom back onto the metal plate.
Are you still with me? Anyway, the chair back is now rigidly attached to the metal underneath the chair bottom, problem solved. Never will I be bothered with chair back wobble any more, what a great advancement for civilization. Of course, I did this to help my back, but I may have done it permanent damage in the process.