I connect all of my BT devices through Settings/ Bluetooth, always, on all of my phones, tablets, etc...
My BT headset, is working as expected, without the help of the Gadgets app.
Sony SBH54 "Smart BT" receiver.
Text messages are read out, other notifications come through, music controls work, I can invoke Cortana with a press and hold on the Call button.
I am not sure what is missing without "Gadgets"?
When I saw this thread I checked the app and found it offered the same additional functionality my Sony app offered on Android.
But when I rechecked on my Android, some of the functions have been removed.
Like, perform an action (open an app, read the time, etc...) when the BT unit connects/ disconnects. It was buggy and didn't always work.
One of the reasons I stuck with Windows Phone was for the native(?) BT funtionality, specifically, reading text over BT. I can't seem to keep Android doing it as consistently/ flawlessly as WP.