A phone is NOT necessary for Auto-Lock
If you unlock your vehicle with a key card, walk-away door lock will not automatically lock the vehicle.
Tesla even implemented a warning message in 2020.36 that informs you it will not automatically lock when you haven't used phone-key: Tesla 2020.36 Update Includes New Phone Key Disconnected Warning - TeslaNorth.com
If your Tesla has 2020.36 installed, a new feature now informs you when your Phone key is disconnected, warning you “Walk-Away lock unavailable”, according to the message.
“Phone key disconnected – lock manually”, reads the alert