I used the .onion link using Onion Browser on iOS. The front page loaded fine, but articles were paywalled. This leads me to believe that it is not browsing through Tor that is allowing the paywall bypass, but rather something Tor Browser is doing.
One experiment might be to go to https://nytimes.com using Tor Browser and see if it also still bypasses the paywall. If my theory is correct, it just might…
The issue you are experiencing likely has nothing to do with the VPN. Network connectivity is not needed to unlock the car. I have been in places with no cell phone signal and it still works.
I do sometimes experience the same issue you are. If I wake up my phone, then it works. So it may be working for you not because you disabled the VPN, but because you woke up your phone and it then sent out the bluetooth signal to let the car know you were nearby.