Why would you need a cleanroom approach? Linux isn't proprietary. It doesn't implement trade secrets. You only have to worry about accusations of plagiarism, effectively.
The reason BIOSes and such needed cleanroom reimplementations was due to the fact they were covered by copyright *and* trade secret laws. You need independent reinvention to prove you didn't steal trade secrets, and to argue against copyright infringement in cases where the resulting code ended up being the same.
But, since you have the GPL code right there, you can ensure your reimplementation isn't the same as the original except for the most trivial subsets of code. And you don't have to worry about trade secrets. There are none in publicly available code by definition.