The ingenuity of the Linnean system of classification resides in that it fuses two orthogonal aspects consistently. This property may be difficult to understand, but here is the simplest explanation I can produce.
Classification is fundamentally dichotomous in splitting groups into groups, splitting these groups into groups, splitting these groups into groups, and so on. It means that classification has fundamentally the same structure as a dichotomous branching process (like a cell line), that is, is the same kind of process.
This fact means that if we want to classify a dichotomous branched process, then we can avoid contradictions only by either choosing a totally symmetrical classification or using an orthogonal system of classification like the Linnean system, because the classification actually incorporates two aspects of the process, ie, in time and over time, and only by these two options can we fuse these two aspects consistently. The problem is thus actually about fusing three dichotomous patterns into one without contradictions.
This problem thus has a (single) solution only if the branched process is totally symmetrical, ie, all branches have branched simultaneously during the process. For all other alternatives, consistency requires an orthogonal system of classification like the Linnean system. It means that asymmetrical branched processes simply lack unambiguous classifications.
The Linnean system of classification thus liberates us from a vain search for single classifications of asymmetrical branched (or nested branched) processes (like cladistics) by making consistency possible. The alternative is thus an indefinite continuation in a vain search to “define the indefinable” (like cladistics), as Charles Darwin expressed it.