How do I play through bluetooth headphones?

It looks like the underlying problem was similar to the one for this question:

https://stackoverflow.com/questions/31380594/linux-bluez-dbus-communication

Except that I'm running a systemd linux setup so I had to do as the comment on that answer suggested:

With newer Systemd-based systems, copy /lib/systemd/system/bluetooth.service to /etc/systemd/system/bluetooth.service and change the ExecStart line to ExecStart=/usr/lib/bluetooth/bluetoothd -E.

– Arto Bendiken

So I did that, restarted, and then started bluetoothctl, and connected manually:

[Bose QuietComfort 35]# connect 04:52:C7:32:68:16 
Attempting to connect to 04:52:C7:32:68:16
[CHG] Device 04:52:C7:32:68:16 Paired: yes
Connection successful

This time when I paired, my headphones recognised the connection (they didn't previously, even though GNOME seemed to think the pairing had worked) and the output device appeared in my devices list.