The reason this happens is it’s because a part of the BLE protocol. The tile is in what’s called a ‘peripheral’ mode and your phone is in ‘central’ mode. Centrals can have multiple simultaneous connections to peripherals, but peripherals only usually have one.
A peripheral like your Tile will ‘advertise’ its services, basically sending a broadcast for anyone to hear, when it’s not connected to anything. But as soon as it connects to something, like your phone, it will then stop advertising. Because it’s connected. So that explains why you see it when you turn Bluetooth off your phone. If you don’t plan to use Tile with your phone and only want it for detection, you could forget it from the Bluetooth settings of the phone to not pair with it anymore, and then you could keep seeing the advertisements.