If a module contains more than one song, you need to either explicitely set the song you want to play using openmpt::module::select_subsong with a parameter in the range 0...openmpt::module::get_num_subsongs(), or with the parameter -1 which plays all subsongs consecutively. Did you do that? libopenmpt generally does not care if there are "---" items between subsongs or not.

Note that all the subsongs in the example file you provided start with four silent patterns. libopenmpt does currently not strip any leading or trailing silence from modules, so you will have to skip about 20 seconds into the subsong to actually hear anything.
You could implement your own stripping mechanism by checking if the beginning of libopenmpt's decoded audio consists of nothing but 0 samples, and then decode another block until you find any audio data that is not 0.

Untested pseudocode which could achieve something like that:

bool first_decoded_block = true; // this needs to be some sort of global variable that is stored alongside the openmpt::module instance
...
do
{
    size_t num_rendered_frames = module.read(...)
    if(first_decoded_block)
    {
        for(size_t i = 0; i < num_rendered_frames * 2) // assuming stereo
        {
            if(buffer[i] != 0)
            {
                first_decoded_block = false;
                break;
            }
        }
    }
} while(first_decoded_block);

Many thanks for your input. I understand the issue now.
I guess the developer of the app I am using would rather like to have this "fixed" in the library itself but anyway I will get back to him with this workaround.

Silence trimming will probably not be implemented very soon, and once it's implemented it will be an optional feature.