A completely different approach said to work better would be to do the opposite. - seal the attic completely and use spray foam under the roof decking and on floor of the attic

You certainly could make the attic a sealed conditioned attic and eliminate the issue, although the cost will give you pause. For a vented attic, I've found that the best venting strategy is a combination of soffit vents and ridge vents. For hip roofs, you can install shingle-over ridge vents at the hips as well as the ridge. It's less obtrusive than turbine-type vents.

Are there no soffit vents? If so then that might be the solution. Cooler air has to enter the attic in order for the hot air to leave. The one gable vent does not appear to be providing enough intake. A temperature of 166 degrees is really hot. Adding more turbines is would be good providing the intake is also increased.

For turbines I'd put them at 1 or 2, 3 or 4 and 5 or 6. Ridge vents would probably work better but you have to take off the original ridge cap, cut through the roof and install the ridge vent caps.

"The one gable vent does not appear to be providing enough intake."

Gable vents contribute little to attic ventilation compared with soffit and ridge vents. In fact, where soffit and ridge vents are installed along with gable end vents, the gable end vents were found to interfere with air flow according to research at the University of Illinois at Urban-Champaign.

I've read about undesirable consequences of "powered" attic exhaust systems. As I recall the problem caused by all such systems, whether turbines producing an exhaust draft by spinning or those using electricity, is that they create a negative pressure condition in the attic.

Yes, that's what pulls in air from the soffit vents. It often also pulls in air from the living space below because most houses are old enough or built in such a way that little attention was paid in construction to having an effective air seal between the living space and the attic above. That produces a negative pressure in the living space and that pulls in outside air. Which, during cooling season in hot and especially humid areas, works against what the AC is trying to do.

I'd follow up on CR Homes suggestion and consider ridge vents. Soffit vents or their equivalent are always essential, to allow cooler air to enter replacing the hot air that exited at the top.

(By equivalent, some houses have attic intake vents that aren't in a closed-in soffits. In my area, most commonly seen is an un-closed area in the two legged open triangle caused by the roof overhang, where it extends beyond the outside wall. Just under the roof plane at the top of the wall surface one typically finds vent holes from the outside to the attic. They usually have screens or grills to keep out larger critters. These serve exactly the same purpose as if the roof overhang space were closed by a soffit which itself had vent openings in it)