Go ahead and fix it to your taste. I don't have anything against the concept of cladistics, but the current practice violates a principle that it took a lot of painful experience to absorb. Namely, if the practitioners of an art can't/won't speak in clear, simple language, that's usually a symptom of massive confusion on their part. I do think the subject deserves an entry. DJK
I've seen some cladistics and as far as I can tell it uses no more jargon then any other field. A lot of the terms, as stated, are used by other methodologists as well. But more to the point, except for perhaps clade each of the words expresses a concept which doesn't really have a compact synonym in normal English. So they're somewhat unavoidable, just like jargon like endoplasmic reticulum is unavoidable in protistology.
Ah, one more thing I noticed. Cladistics is not a classification system, as stated on [Linnaean Taxonomy/Talk]?. It's a methodology for elucidating evolutionary relationships. Taxonomy ties in because cladists intend it to reflect the phylogeny of organisms. However this intention is shared by pretty much the majority of biologists. If you look at some of the more recent Linnaean schemes for, say, protists and flowers, you'd find everything up in the air for precisely these reasons. So objections to the practice on these grounds have kind of been superceded.
On these grounds I think I'm going to take a stab at rewriting the article. Please don't be insulted by this! I'll try and leave all your points except the above, but move them to a criticisms paragraph down at the bottom, since that seems to be the standard way of presenting a neutral point of view. Personally I think cladistics is neat, at least when it works, but takes forever to get used to. :)