Front end shimmy
 

The wife's 190E has developed a fairly rough shimmy at around 55 through 60 mph. It goes away at higher speeds.
Is this more likely a tire out of balance or a bad tie rod?

I would start with the tires.

Tires, tie rod ends, ball joints, control arm bushings. They all wear out and loosen up. An easy check is to get under the front of the car so you can see the tie rod ends. Have the wife roll the steering wheel left and right about 180 deg. repetedly - about 1sec. in each direction. Worn tie rod ends will sometimes "pop" up a little each time the direction changes. Also you can wrap a rag, single layered, around the ends and hang on to each one while they are moving side to side. You should not feel any clicking or popping. If you can see movement, I'd replace them.

You can use this test to generally observe the condition of the joints too. You shouldn't see them "working" up and down or moving side to side in relation to themselves. This is to say that there should be no observable radial or axial play for lack of a better description.

I go with all of the above.

It is likely an out of balance tire, or an out of true wheel/tire.

As the car ages, struts and joints get loose, a "shimmy" from an out of balance/true wheel will become more severe. So the root cause is probably the wheel/tire, but it might be showing some loose suspension parts.

When was the last time the power steering fluid was changed? If all of the above doesn't work, flush/change the fluid. I had an impossible-to-kill shimmy in my W140 that disappeared after a PS flush & fluid change.