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I thought even today phytoplankton produced more oxygen than all the multicellular plants put together.

I might just be imagining that factoid though.

That was my understanding as well. The idea being that there is a lot of volume in the first few meters of ocean surface area churn. It's not hard to imagine that does better in surface area than (even a lot) of plants.

Not forgetting the fact that it seems like it would be substantially easier for a faintly mobile fluid filled bag(s) of DNA to survive and reproduce in an aqueous environment!

You're getting ahead of yourself a bit. By a billion or two years.

The lifeforms that oxygenated the atmosphere were rather more primitive than trees. Generally cyanobacteria. Technically, yes, plants.

Most oxygen (even today) is produced by single-celled algae in the ocean.

They're certainly sufficient but not necessary. There's no reason why single celled organisms couldn't do that. Maybe not at 20% but close.