I find it hard to believe that it's impossible to make a glibc competitor that is LGPL and uses kernel headers. Could you point me to some opinions that you're basing this on?
During all those years that Linux has been around, no one other than Google has tried to take this kind of material from the original kernel headers and publish it under any license other than the GPL. And no one has tried to do an alternative glibc. I guess what someone would do is take glibc and fork it on LGPL terms.
You can build libraries if you like, but you can't just put header code under a different license unless you're the original right holder or have the original right holder's consent.