I've now tried simply connecting my phone to this AUX-In connector with two different mini-jack to mini-jack cables.
The phone recognised that the "headphones" are plugged in, but nothing is heard on the car speakers whatsoever when I start Spotify for example.
I've changed the option to AUX on my head unit.
The only thing I can think of that may be causing a problem is that I'm not using the stock stereo, but an aftermarket Pioneer one that was professionally fitted and works fine with all the steering wheel buttons etc..