Well said, and, other than the "Mobile" version (which I have no interest in, so haven't researched), they are already out there in use.
For instance, this is my current life (and this is by NO MEANS comprehensive or complete, or even possibly fit for others!!!):
Server - CentOS: no GUI by default, many choices during install for different server functions, dev and build libs easily available, no real bloat, lean and fast.
Desktop - Mint, Elementary, etc... Each are user-friendly FOR THE USER, including those migrating from Winblows or Mac.
I'm a sysadmin, have been for 20 years, and have always used some form of Linux and as few M$ servers as possible. And we're successful and have a growing body of customers using our servers and services.
HOWEVER, I agree with you in that a distro should firmly recognize and position itself into one of these three flavors (using your syntax), and not try to be all three. It would make selection and use more clear and easy. It would also help those distros become better at what they are trying to accomplish, rather than a "one ring for all" mentality where nothing works as well as it could and should.
That's why I chose CentOS for my servers (as their mentality is SERVER, not desktop or GUI or some other function) and Mint or Ele for desktop (again, they are focusing on an end-user experience, user application stability, and usability, rather than server functionality). I am NOT saying this is right for everyone, just what I have found works perfectly for ME and MY OPERATION. CentOS may not work for some server applications (though, I haven't found that to be true), or there may be something about it that another sysadmin may hate... That's fine, there are other choices. That's the great thing about the Linux universe.
And Mobile fits right in there too: a moble OS GUI should be VERY different from a desktop OS GUI, as Micro$oft has so eloquently proven. No suggestions from me. Though, I carry an Android phone, if anyone cares. :)
The right tool for the job. The Linux distros should embrace this concept, and if it would, the world would be a better place.