The answer to your re-phrased question
"Do any chemical processes occurring within the physical volume of any entiy considered "living" depend in any way, shape, or form on oxygen's paramagnetic properties?
is an emphatic 'yes', and this answer may be substantiated with a single word: haemoglobin.
Pauling and Coryell discovered in 1936 that the oxygen in both oxy-haemoglobin and CO-haemoglobin is diamagnetic, and I quote from the last paragraph of that great paper:
It is shown by magnetic measurements that oxyhemoglobin and carbon-monoxyhemoglobin contain no unpaired electrons; the oxygen molecule, with two unpaired electrons in the free state, accordingly undergoes a profound change in electronic structure on attachment to hemoglobin.
This result has 'stood the test of time' and is a key observation in explaining at a molecular level the allosteric and cooperative properties of haemoglobin. IMO, it is one of Pauling's great observations, and illustrates his phenomenal powers of deduction.
As Pauling points out, Faraday also investigated the magnetic properties of haemoglobin and recorded in his notebook 'Must try recent fluid blood'.
To again quote Pauling and Coryell :
If he had determined the magnetic susceptibilities of arterial and venous blood, he would have found them to differ by a large amount (as much as twenty per cent for completely oxygenated and completely deoxygenated blood); this discovery without doubt would have excited much interest and would have influenced appreciably the course of research on blood and hemoglobin