Well, this should be fun. I've seen the engineering people often fret over designing something that a technician would not assemble the way they want it to be done and risk a problem here on this site as well as on the job. Through-out my career as a technician I have had to deal with situations where everyone else simply throws guesses, and treats "us" as if we have no real talents.
The "flat spotted tires" is essentially a guess Pat. While it may be somewhat likely in some cases, simple details about how the reported problem was written actually rule it out. The problem from a techs standpoint is that a tech could replace the tires, and then when the problem is still there (as it is) now the tech and the repair shop are often expected to find and fix the noise, for free! The shop could also be expected in many cases to give the money back that the customer spent on the tires. I can go into greater detail later, but both of these scenarios work to drive potential great automotive technicians, who could solve goodboats problem into other fields.
From Goodboats original post.
"What might be the cause of a periodic low-frequency vibration in a car. The vibrations sound a bit like muffled wind-buffeting, on the edge of being unnoticed, and pausing every second or so."
At what speed does this occur, and what speed ranges will this definitely NOT occur?
Does turning either by going around a bend, or simply swerving inside your lane have any impact on the sound as it is occurring? (lessen, or increase as compared to turning one direction over the other)
Does surface wind direction/speed play a role in the noise?
"The frequency does not change;'
This line right here rules out tires.....
"the vibration just comes and goes. Can be felt, just barely, in the steering wheel, but the sound seems to come from the whole body of the car."
Seriously, if "the entire body of the car" seems to be involved in the noise, then it would make sense that it can be felt with your hands through the wheel. So, lets concentrate on the entire body, and NOT the steering.
"Seems more noticeable when car and tires are very cold. Poor tires increase it, new Pirellis lessened it."
I call this "tainted intuition". In the shop this kind of information would not help me diagnose the problem. In fact, it easily could lead to to looking at tires, when they should already be ruled out!
" Have had this '99 Mercedes 4-Matic wagon from 82,000 miles to 126,000 miles."
The noise, or sensation your experiencing, when did it start?
"Alignment has been checked repeatedly, 4-wheel alignment (supposedly)."
More tainted intuition. Concentrate on the problem right now. What's been done in the past has absolutely no bearing on the situation.
"This car has full-time AWD but different percentages of power to front and rear, 40 vs 60 percent, I believe."
Not relevant to diagnosing the problem at this time.
"If I take the car to a dealer and ask this question, I'm always told the mechanic can't hear or feel the vibration."
No surprise there. How have you set the stage for them to experience it? Right now I could ask you if you could tell me if we drove from YOUR HOUSE, to a specific location would we definitely feel the problem? Your answer could easily be yes, and one of the possible reasons for that is the fault creating the vibration could actually be the surface of road your regularly driving on. Don't laugh, or attempt to deny it because that has been a real diagnosis many times! What we cannot do is simply guess (like the flat spotted tires). We have to spend time and investigate the issue. Now, one might say if its the road why doesn't the frequency change? The answer there is simple, speed limits, and the fact that no one has investigated this possibility yet. Until someone has, then its in play.
Now a few last questions. Do different people inside the car experience the symptom the same or differently? In other words, if your son drives, and you ride can you move around to different seats inside the car in an attempt to get a source direction? What about adding passengers, and/or changing tire pressures? How does that impact the issue? If you cannot force a change in the issue, by any of these, steering, passenger weight, speed, etc. Then the problem is NOT tire, road related. Now you have to actively be looking at things like plastic air dams that control airflow through the radiator (if the problem is in the front of the car). The plastic shield that is connected to the body under the engine is one such panel.
So before you go back to the shop, again.. Lets see how well the vibration can be observed and then explained here. That way you practice painting a clear picture of exactly what I would expect to find when I drive your car to experience the problem. Be prepared once we reach that point, that you may have to ride with the technician to show him/her the problem. One thing, use hand signals to show when you feel it, and don't feel it. Trying to say "there, did you hear that" will only be an interruption to the techs observation.