That's strange. My Surface Pro 3 lets me play music and even press the power button to force it into sleep mode (screen off, fans off) yet the music keeps playing. Basically just like my Windows Phone does.
Microsoft solved this specific problem (impossibility to turn off screen) with small trick. Native apps can hold device in active mode althought you press a power button. On my device it is working good an there is not any time of limitation. Music will play constantly but I discovered one difference. Gap between tracks takes 5-6 secs. Next app wihich cant working continuosly, indenpendent to settings in power plan, is TuneIn radio.
I thing that only way how to beat Microsofts stubbornness is using some tricks. Its sad. :(