The ocean breathes in and out too. Very little carbon sinks down to the sea floor. The ocean does happen to absorb a lot of CO2, but that's from acidification rather than photosynthesis.
Yes agreed. But the organisms breath "out" much more oxygen than anything else. And yes ocean acidification is both what's driving a massive crisis and saving us from the full impacts of our CO2 output. It's not a pretty picture
You could induce artificial algea blooms. All that needs is iron and light + water soluble co2 and / or o2. If you are greedy you pump light down into the depths , go cubic instead of quadratic?
Intentional oceanic iron fertilization (OIF) is actually well-studied. Unfortunately it carries risks of ecological disruption and can lead to deoxygination (dead zones). Which would long-term hinder our ability to repeat this. It also often leads to methane and nitrous oxide emissions which are both very potent (though shorter lived) greenhouse gasses