From how I learned it, the kernel is the mainfray of the system. It is what makes the OS Linux. And while all the other tools sit atop and interact with the operating system in a symbiotic relationship, it is the kernel that effectively defines the operating system.
My point is that with the above in mind, the Linux kernel is the Linux system as it strictly defines the OS. The GNU overland tools can be run in some form on other kernels (e.g. the BSD's, Darwin/Mac OS X, Solaris, Unix, GNU Hurd, etc.)
However, you're right in that there's a problem. One can only compile the Linux kernel using gcc. And usually there has to be some *nix system running.