Currently, only when you press the volume hardware keys when you are in the view of the media control they work and trigger the connected outout device.

Actually, IMHO we can expand this, as I see a common use case:
* you usually want to hear the music you control by yourself, i.e. e.g. have headphones connected to desktop
* you usually also only control one such audio output device (makes sense with first point, as you are unlikely to "mix" it with different devices.)
* also in this case, you obviously do not hear any music on your phone itself, then you would not need KDE Connect

So if:
* only one device is running music
* a special setting is enabled

KDE Connect app should IMHO enable "volume key" overwrite globally.

It would be great to also have that feature on the lock screen, because you can then also control the volume without unlocking your phone. :)