I really like to learn even a single case that this ancestry-focused classification system is useful to solve a practical question.
You can start here:
Neighbor joining
What I meant is that if we do not involve the use of ancestry, then this classification system won't work in a practical sense.
Like I said as long you assume their exists a world wide conspiracy that somehow secretly rigged every single cluster algorithm is such way as to prevent loops to be created when investigating the
distance matrix, this without anyone noticed this, with the purpose to make it look like common ancestor exists then you will learn noting.
If you assume this, then I challenge you to show us where this rigging occurred in the algorithms so we can use the proper cluster methods on the distance matrix (and by the way you can also tell use how evolutionist Carl von Linne rigged his classification to create the same relations as current algorithms produces).
If you cannot do that then next step is to investigate if the evolutionist conspired to rig every single of the distance algorithm uses, for instance where did the pesky evolutionist
Euclides assumed a common ancestor when he came up with his famous distance formula:
A^2 = B^2 + C^2
Because what you actually say when you claim that common ancestor is assumed in cladistics is that the evolutionist Euclides assume a common ancestor in order to be able calculate the length of the diagonal in a triangle.
However, it is not obvious how he did this, so his assumption must somehow be very cleverly hidden in the formula because nobody, for well over two thousands years, has yet seen it.
This is your chance! You may here go to history as the first creationist who uncovered the evolutionists worldwide conspiracy and showed how the theory of evolution was all based on a false assumptions of an common ancestor.
Not to mention that you also will be the person that made the revolutionary discovery in mathematics that 1+1 cannot possible be equal to 2 unless a common ancestor is assumed for all life on Earth.
If none of all this silliness is your assumptions (and I guess it is not if you are at least half as sane I suspect you are) then I challenge you to come up with a clustering algorithm that when apply on the distance matrix will not produce a nested hierarchy. If you cannot do that, then you must accept the fact that shared common ancestor is an observational fact - or continue deny the facts (which I suspect you will do).