Well, the problem was really behind the keyboard.
So the publishing profile is actually provisioned correctly and automatically to the Web project as the documentation states.
The problem and confusion was that I have a secondary web application in my Solution that I also publish to the same Web Role (referenced as an additional Site
in the ServiceDefinition.csdef
file).
That Web Project does not get the Publishing Profile, and when I try to create a profile manually, it doesn't work since that (secondary) IIs instance is not configured for Web Deployment.
Oh well, back to the tedious Cloud Service deployment it is...