The difference between Linnean systematicians and cladists is just between believing that there is a single true classification of life or not – Linnean systematicians believing (or understanding) that there isn’t and cladists believing that there is.
This difference makes no difference whatsoever on whether there is a single true classification or not (there provably isn’t), but just splits biological systematicians between non-believers (traditionally called atheists), ie, Linnean systematicians, and believers (traditionally called extremists), ie, cladists.
This split is extremely interesting in that cladists think they advocate science against religion, when they actually advocate one belief against other beliefs. They have misunderstood science as assuming that kinds are real, when science actually just use kinds to discuss reality. Their interpretation of science does thus place science as a religion among other religions, when science actually (see Karl Popper) is the opposite to religion. They have thus misunderstood science fundamentally and instead emerged as the unwanted extremists of science.
Science’s worst problem is to handle extremists of this kind (ie, cladists). The general question is: how do we handle people that believe what we say?