Truing tires. Is the tire properly mounted? Is the tire even to cause of vibration?

Had a question from "Fred" on "Truing Tires"
On a different subject, on the Escapee's forum, there was a conversation about tires, and someone mentioned truing new tires. IMO, truing of new tires today only covers up poor mounting procedure, vs 40-60 years ago when tire construction was not as advanced. What are your thoughts on this?

 First, let's be sure everyone knows what we are talking about.
The process of "Truing " a tire involves shaving part of the tread rubber off to make the mounted tire more round. HERE is a YouTube video showing the process.
I tend to agree with "Fred" that most out-of-round problems can be traced to a tire not being properly mounted and seated on the rim. It is also possible that the wheel itself may not have been properly mounted on the hub or it is even possible for the wheel itself to not be round.
 If you have a vibration problem you first need to ask "What has changed?" Did you just get new tires? Are the tires the same but the wheels were off the vehicle for something like a brake job? Have you been driving on especially rough roads with many large potholes? This is a "new to you" vehicle and you have no history with the tires and wheels on this vehicle. On some vehicles "Heat Set Flat-Spot" may be the cause. This happens when tires have been run, getting hot and you just stop and park the vehicle. Sometimes leaving the vehicle parked for weeks or months can also allow the tire to get a flat spot which could take a couple of hours running to work the flat spot out. High-Performance Passenger tires are more susceptible to this problem than large 22.5" size radials but even 22.5 tires can develop a flat spot.

The problem of vibration can usually be traced to either out of balance or the tire/wheel assembly being "Out-Of-round".
If you have vibration on a new tire, I would first confirm it was balanced. Next, I would measure the assembly on the vehicle and confirm it is in-tolerance. This usually means less than 0.030" radial runout goal with 0.125" the upper limit.
If you are exceeding the above, the next step would be to try and confirm which component is not "round". Ideally, you would confirm the wheel, with no tire mounted, was below the limit. If the wheel is "round" when measured on a mounting machine then we would need to confirm it is round when bolted to the hub. Some wheels are "Hub-Centric" and others are "Lug-Centric". Hub centric means the wheel centers on the ledge of the hub on the brake drum. Lug centric means it centers on the bolts holding the wheel to the hub. A lug-centric wheel can easily get "off-center" if one lug nut is fully tightened before the other lug nuts are snugged up. There are patterns for the sequence and steps of tightening lug nuts depending on the number of lug nuts. Here is the sequence as published by Chilton, a publisher of numerous automotive repair manuals.

 
 In addition to the above sequence, it is advisable to tighten the nuts in three steps of 1/3rd of the torque level. Example: if your torque spec was 90 Ft-Lbs you would first do all the nuts to 30 ft-lbs then again following the sequence tighten to 60 Ft-Lbs. The finally to the 90 Ft lb spec, again following the pattern
If the pattern and amount of torque are not followed it is easy to end up with a round tire & wheel assembly to be mounted off-center to the hub which results in an out of the round situation and unacceptable vibration.

Before I resorted to shaving a new tire I would measure the out of round. If you have confirmed all the above yet still have vibration on new tires, I would work with the tire dealer to confirm there is nothing in the individual tire that might contribute to some vibration by simply switching tires around or trying a different set of tires.
In all probability, by now you would have found and fixed the cause of the vibration.