Thanks for the question. It got me digging. I think the reason you probably get silence or deflection is because it isn't as simple as a quick google search or parsing through just one case. The principal doesn't rest on a case decision and isn't explicitly forbidden or "so ordered" as most of what we talk about in this sub is.

The best single case id Indianapolis v. Edmond where SCOTUS threw out the ability to have "Drug Enforcement" checkpoints because the checkpoint program's primary purpose is indistinguishable from the general interest in crime control, the checkpoint therefore violates the Fourth Amendment.

I would suggest reading the entire opinion as well as the petition for Indiana v. Edmonds. The way it is pieced together in many ways implies that checkpoints each rest on their underlying case, Prouse for "information," Sitz for DUI and Martinez-Fuerte for immigration.

The "Primary Purpose Principle" rests on the Indianapolis v. Edmonds case...

Finally, we caution that the purpose inquiry in this context is to be conducted only at the programmatic level and is not an invitation to probe the minds of individual officers acting at the scene.

Which effectively means that the purpose of the checkpoint has to be outlined (not necessarily in writing) and not simply administered by the officer on site.

I have started parsing through a search for "Primary Purpose rule checkpoints." I would recommend doing the same, it looks like there are many good sources and maybe you can find more than I did so we can get a clearer picture, but here:

What steps may an officer take to detect motor vehicle violations at a checkpoint?

There is no single laundry list of permitted investigative techniques. Generally, an officer’s actions must be “reasonably related . . . to the circumstances which justified” the checkpoint. Therefore, an officer may certainly ask to see a driver’s license and registration at a license and registration checkpoint, although it is unclear whether an officer may conduct a computer check regarding an apparently valid driver’s license. Likewise, at a sobriety checkpoint, an officer may attempt to detect impaired drivers using nonintrusive techniques like those described in State v. Colbert, where officers “observed the driver’s eyes for signs of impairment, engaged the driver in conversation to determine if the driver had an odor of alcohol on his or her breath or if his or her speech pattern indicated the possibility of impairment, and observed the driver’s clothing.”

Source: UNC School of Government