Hello Raphael,
I just tried some to perform a classification using neural network and I encountered the same problem (all samples are classified as the first class). I think there is something wrong going on here. I need to investigate.
I the meantime you can try to use an older version of OTB, for example OTB-6.6.1. I managed to train a valid neural network classifier with this version and it should be compatible with QGIS. You can find it on the archive page : Index of /packages/archives/OTB
Cédric