What do you mean by 'independent toolchain'? And what part of the OS were you looking to learn about?
Remember that LFS, despite the name, is not about learning Linux-the-kernel, it's about learning 'Linux-the-operating-system' (which is more precisely referred to as GNU - using 'Linux' to refer to the OS is technically incorrect, though very widespread).
Most of the time this is just a point of either pedantry or politics, but if you want to build your OS from scratch, you really need the entire toolchain[1] if you want to build a 'usable' system.
[1] You can sub parts of other toolchains, so I guess the more correct answer is that you need an entire toolchain, with the GNU tools being the de-facto default toolchain.
Agreed the toolchain is a requirement. But you raise the problem, what LFS teaches you Linux or 'From Scratch' ? I[1] didn't learn about the kernel nor the operating system there. Only about package building. The OS remain a mystery to me for most of its part. See my example about elinks problems.
Remember that LFS, despite the name, is not about learning Linux-the-kernel, it's about learning 'Linux-the-operating-system' (which is more precisely referred to as GNU - using 'Linux' to refer to the OS is technically incorrect, though very widespread).
Most of the time this is just a point of either pedantry or politics, but if you want to build your OS from scratch, you really need the entire toolchain[1] if you want to build a 'usable' system.
[1] You can sub parts of other toolchains, so I guess the more correct answer is that you need an entire toolchain, with the GNU tools being the de-facto default toolchain.