That is correct, there isn't an ActiveSync equivalent, you would simply create an app that detect and copy data from the SD card within the device. As you are probably aware, connect the device to a PC and you will be prompted to establish a connection. The process essentially dismounts the SD card from the device and shows as an external USB drive on your PC.
Alternatively, you can interface with Android devices through ADB (Android Debug). Check out page 99 from the ET1 Integrator Guide, Section: ADB USB Setup.
Let us know how the Xamarin approach went.
After more than 3 years, Is there any other solutions ?
I doubt there will ever be a traditional ActiveSync replacement as everything is via the cloud now. There are innumerable cloud file sync solutions for both GMS and non-GMS android devices (e.g. FireBase works well) or if devices are being managed by an enterprise mobility management (EMM) provider many of these will provide device content management.
And as Peter says there's always adb to physically connect to the device and pull files off.