So I now have exactly this problem too. I'm using exactly the same method of connecting through the browser as I have before, and all I get is the orange bar and "Device connection failed,".
I did the reboot and the Bangles shows '-> Bluetooth', however the BT widget is grey, the phone (to which it's usually connected, and the only device other than the browser to which it's ever been connected) has BT switched off - so it's not connected to any other BT. I have forgotten the iPhone pairing, but can't find a way to forget the browser pairing. (Update: forgot the pairing in the System Preferences instead of the browser.)
As I say, I've done nothing different from what I always did (phone BT off, connect through the browser), but can't get Bangle connected in the browser. It does connect just fine with the phone.
UPDATE: Solved the issue, but not sure how. I had deleted the pairing from both phone and computer. When I tried to repair with the computer, it asked for a "passcode shown on remote Bluetooth device" but there was no indication of a code on the Bangle. Without closing that window, I opened a new window with the Apps loader page, and it connected.
I tried to repeat this connection method, but it has failed again. Still getting the orange "Device connection failed," bar. Clicking Cancel returns to the orange bar.
I've never had this connection issue before.
UPDATE UPDATE: In order to be able to connect, I have to hold the button in through the <====> screen, until it boots to the logo and just ">" bottom left. Then I can click Connect on the App Loader page and it connects.