Cladistics only recognizes what it callas clades, which it defines as “an ancestor and all its descendants”. This definition is, however, the traditional definition of a clone.
So, is there any problems with only recognizing clones? Well, the first problem is that clones is not a general, but a specific concept, ie, is a clone of something. Not all living beings form clones. Me, for example, is not a part of a clone. This problem does cladistics, however, avoid by discarding individual organisms, instead only considering the two kinds of things it calls populations and species, whereof it doesn’t define population, but defines species as “a lineage of populations between two phylogenetic branch points (or speciation events)”. It thus, in practice, defines species as “a lineage between two speciation events”, wherein lineage is undefined, and thus actually defines species as “something between two speciation events”.
This definitional trick does indeed create an abstraction called species, ie, “something between two speciation events”, but does at the same time contradict the clones cladistics calls clades in that such clades contain a single first and many last things which are not “something between two speciation events” (ie, are not species). The trick is thus incompatible with the definition of clades.
The overall picture is that the fact that clones is a specific, not a generic, concept means that it can’t be combined consistently with a generic concept, like species. It does thus not matter how hard cladists try, they will never produce consistent definitions of both clade and species. This is just as impossible as to decide both the position and momentum of a particle at the same time. Cladistics thus tries to solve the insoluble, and its solution is thus always consistently contradictory (actually paradoxically contradictory), whatever it is.