How do I reconcile this with "entropy invariably increases" which is a contradiction to your hypothesis that "nature tends to favor zero-sum games"?

How do I reconcile "for every chemical and nuclear reaction, when something is gained, something else is lost" with catalysts increasing rate but not being consumed themselves?

In fact you can show there are an uncountably infinite number of broken symmetries in nature, so it is mathematically possible to concoct a parallel number of cases where nature does not have some "zero sum game" by Noether's Theorem.

Your statement is just cherry picking a few and then (uncountably infinitely) overgeneralizing.

You are picking apart the zero-sum part, I would also pick at the 'game' part. None of the examples are 'games'. They are just a few examples of conservation laws.

No decision making, no min-maxing actors etc.

Btw, when you have a single optimising actor, then moving along the efficiency barrier is also a set of trade-offs, which can be made constant-sum, if you set up your conversion factors just right; even if the optimisation itself is otherwise variable sum. (As a silly illustration: to produce more guns, you need to produce less butter.) But that observation doesn't really prove anything.

Entropy is a measurement of the system itself and doesn’t describe the dynamics within that system.

Catalysts increase reaction rate just as a train runs faster on a track. Is a railway a catalyst?

Are symmetries broken in nature or just models of nature? Or are you referring to accepted theories in theoretical physics, which was the entire point here?