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You wrote:
<I think cladistics _removes_ much subjectivity. Not all, but lots. No more
arguments about what rank a particular group should be given. No more
paraphyletic groups. Parsimony. Etc.>
and answered yourself:
<The first two arguments apply of course to phylogenetic taxonomy alone and
not to cladistic analysis. My error. But since cladistic analyses are
reproducible, testable/falsifiable, and make predictions, they certainly
remove subjectivity and add science to what used to be an art.>

Terminology:  repeatable rather than reproducible, please.  Running the same
program on the same data a second time is a repetition.
Also, cladistic analyses generate hypotheses.  When you have a hypothesis
about extinct animals, you cannot immediately use the standard test of
making predictions from the implications of your hypothesis because you
cannot rerun evolution.  You can't generate relevant observations at will.
You can, however, predict that a fossil with certain characters will be
found, dating from a given time period.  Then, of course, you can settle in
to wait and see what happens.
(Historic sciences and some physics all have this problem; the search for
confirming observations can be difficult.)

The other problem to keep in mind is that nature is known not to follow
parsimony (or any other logical principle) in every case.  You don't know
that parsimony must give you the correct answer.  Therefore, the use of
parsimony is a choice (albeit a reasonable choice) on the part of the
analyst.  Any given conclusion is no more likely to be correct than a
conclusion based on a different logical principle, unless there is an
argument that parsimony is the most likely operant principle in the specific
circumstances compared to the alternatives.

In short, a more elaborate approach like cladistic analysis is not
necessarily more objective or more accurate than an alternative, less
formulated approach.
Nothing wrong or unreasonable about it, just not the perfect and only
solution to finding objective and true answers in evolutionary biology.
Says me...