Bluetooth is always visible, even when the phone screen is off.
I frequently connect my ear buds on other computers via BT and I always see my phone listed as BT device.
I believe this is not the desired behaviour nor it is safe.
Could you give more details about the issue with your device's Bluetooth?
You mean that your device appear available to connect via Bluetooth while this function is enable or also while it isn't enable? Or you mean that while is paired is visible to other devices to connect via Bluetooth?
I think the issue is Bluetooth remains discoverable once turned no matter what the state of the phone is in. Best behavior is to have Bluetooth on and only discoverable when needed like having a switch that will timeout.
On my device it seems to be working. If not in the Bluetooth settings page, it is not discoverable. Only becomes discoverable once in the settings page is active.