Tbl 1-2-1 in the AIM shows required navigational performance (RNP) required for different phases of flight. Enroute is 2 NM from centerline. Terminal is 1 NM and approach is .3 to 1 NM. Also from the AIM, "For an aircraft to meet the requirements of RNAV, a specified RNAV accuracy must be met 95 percent of the flight time." (Sorry for the AIM vs FAR reference. I don't have my tabbed out and highlight FAR/AIM with me. The AIM is easier to search.)
But that centerline accuracy (my term, not a FAA term) is what is graded by the Check Airman. So there's some standard that indicates a lack of adequate precision. That's what I was getting at with the PTS. I think PTS says full deflection throughout the regime of flight, so for enroute, that goes back to your "4 NM and 1K'" quote, which makes sense, since those numbers pop up every where else. But from the terminal phase all the way through the departure phase (there's a fixed, manual RNP for proprietary departures, as well), the scale changes, so "full deflection" will be less than the total of "4 NM."
None of this is to argue against your point, just expanding on the PTS comment and why it's not a number/XTK per se but an idication (full deflection).
TACAN/VOR generally gets more precise as you get closer to the station (the "cone of confusion" at high altitude nonwithstanding), which is what enables it to be used for non-precision approaches. Many of those approaches require PTPs to reach the IAF, and a proper understanding of geometry and aircraft turn performance in order to arc and turn to radials. So yes, poor execution would be an issue- but you could argue the same for a poorly hand-flown RNAV approach. Bad piloting is bad piloting, regardless of the nav source.
But the GPS scales as you move through the approach, so it's simulating the cone of confusion in the same way, just with significantly less error and more precision than a navaid.
Or maybe I'm misunderstanding what you were getting at.