It could be a combination of both theories. Increased oxygen allowed animals take deeper vertical dives, and these deeper dives created yet more usable oxygen, making even more deeper dives possible. It could be that animals first had to reach a threshold of complexity to trigger the feedback cycle.
For example, there's less need to hide in the depths from predators if the predators are wimpy, and predators without enough oxygen will stay wimpy. The diving energy expenditure is not worth it under wimpy predation. Capability to dive deep enough and the predator/pray competition both had to reach a level to trigger the feedback cycle. Earlier generations couldn't get either the depth nor be potent enough predators because they were too primitive.
Or nitrogen fixing, the oceans aren't great for nitrogen, being made of water rather than air. Oxygen isn't that scarce at the surface levels, normal waves help with that.
Around 40 million years ago when grasses started to take over there was a big resultant change of what went on in the ocean, more nutrients, i.e. fixed nitrogen. This was former 'soil' blown off continents that had gone through a bit of climate change. River sediment too.
What happened at the big bang for life 500 million years ago to resolve the nitrogen problem is beyond my wikipedia skills.
Also what was the original atmosphere made of, before the oxygen takeover?
What I think is amazing is that every rusted or oxidised thing had to get its oxygen from somewhere, i.e. the waste product from micro-organisms getting their carbon. The atmosphere only came about when everything that could be oxidised was oxidised.
Does diffusion of oxygen really need active mixing?
I would have assumed even with a low oxygen atmosphere the oceans would slowly get saturated with oxygen by diffusion and existing currently
Or do they mean that the rate needs to be diffusion higher than at what the animals are consuming it? That's a interesting question. Do animals actually make a dent in the ocean's oxygen content? If their rate of consumption is small enough it probably doesn't matter too much.
I mean, the guy putting forward this theory is an expect in fluid dynamics. So I expect the answer is "yes, animals do make a big impact on oxygenation of the ocean".
Here's another study from 2014 that talks about the link between animals and oxygenation of the oceans. (This paper just focused on the mixing of oxygen in the ocean, it doesn't claim that more ocean-oxygen led to a higher oxygen world like the OP. https://www.sciencedaily.com/releases/2014/03/140309150540.h... )
I feel like this could be proven/disproven mathematically using an average oxygen diffusion rate over time calculation for whatever species the author is referencing.
ex: Microbe A diffuses (0.0003mg of oxygen per day)x(population of species)x(years) = 100 billion tons of oxygen diffused into atmosphere per millenia
For example, there's less need to hide in the depths from predators if the predators are wimpy, and predators without enough oxygen will stay wimpy. The diving energy expenditure is not worth it under wimpy predation. Capability to dive deep enough and the predator/pray competition both had to reach a level to trigger the feedback cycle. Earlier generations couldn't get either the depth nor be potent enough predators because they were too primitive.