Well obviously it isn't a LOGICAL fallacy. It's a fallacy in the sense that you cannot categorize things as living or dead if you do not have agreement on a definition.
It is rather easy to predict that a large animal flattened by a steamroller will not spring back up like a LooneyToon. It is somewhat less certain that a tree is dead when cut down.
But I don't believe the discussion was about legal, clinical death. It was about defining that particular set of criteria that distinguishes life from non-life, in the abstract.
The very fact that we argue about viruses, prions, computer viruses and such indicates there is no clear set of commonly agreed upon criteria. The problem could get much more complex if someone discovers a bootstrap sequence for synthesizing DNA, RNA or proteins.
After this fallacy of quantizing the continuum was asserted, two other models were raised: Irvin Bauer's model which is part math and part characterizations of biological life - and George Javor's which is entirely bio/chemical (and creationist).
If we were to go down the path you suggest (bio/chemical) - then I suspect we will run into a lot of subjective interpretations which will tilt to the ideology one brings to the table. Mathematics on the other hand is objective and also neutral to all ideology and theology.
I'm glad to pursue any of the models in any context (abiogenesis, life principle, fecundity principle, cosmology, geometry). But it would be a duplication to pursue it here instead of on the Plato thread where so much research has already been posted.