Throwing this in your .hadooprc results in hadoop-aws being in the classpath but not hadoop-azure*:
It would seem that the core issue is that that requires the module to have listed it's dependencies in MODULE_NAME.tools-builtin.txt, whereas the Azure connectors only have them listed in MODULE_NAME.tools-optional.txt. S3 does both, and there's a comment in it's POM about how it needs to do this because of the "hadoop s3guard" CLI.
Maybe there's some history that I'm missing here, but I think what's wrong here is that hadoop_add_to_classpath should get what it needs from optional modules. builtin modules shouldn't even need hadoop_add_to_classpath to be added anyway.