None exist without significant privacy concerns

Slate Truck. Extremely stripped-down. No cell radio. No GPS radio. No WiFi or Bluetooth, unless you stick some kind of hardware that can do them in there. Hell, no touchscreen car computer, just a place you can mount a tablet if you want. Doesn't even have automatic windows as a base option. Aside from having airbags, air conditioner, and being an EV, basically a 1980s-style vehicle.

It'll fail some of your other requirements, because it's new and you'd need a used vehicle to hit a $5k price point. Just pointing out that it'll meet the privacy point.

I do agree that it is depressingly rare to find modern cars that don't have privacy issues. It's not that I expected all of them to be privacy friendly, but I expected there to be more options.