Has anyone ever adjusted the steering box per the repair manual's instructions?

and I've also done a few dozen on similar systems on other brands.

So what is the problem???

Sure. Can make a huge difference in steering precision and feel.
Not easy to get at that adjuster, though, at least with the 103 engine...

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1986 300E 5-Speed 240k mi.

A tad too much steering slack!

That might do it, so long as the play in question is ahead of the linkage.
This can be easily tested:
Have an assistant rock the steering wheel back and forth within the 'play' zone, and observe whether or not the motion is transmitted through to the linkage.

In my limited experience, every older 124 benefits from some amount of steering box adjustment. Start with just 1/2 turn, and see how it feels at speed, and in sweeping turns. If you overdo it, the steering will be too heavy, and will bind coming off of center.

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1986 300E 5-Speed 240k mi.

Thanks CS!!!

Remember that to tighten you turn the screw OUT or counter clockwise.