Hugh Laurie, is an actor who plays a doctor on TV, he's NOT someone who should be anybody's opinion leader, except for maybe on how to be a good actor.
Most of the Oxygen is produced by the green algae in our oceans as a waste product from its process of converting CO2, water and sunlight into sugars and plant tissues for growth. It's called photosynthesis.

Randy

As an Ecology teacher one of the most profound revelations I had was that it's not the trees that give us most of our Oxygen - in fact it's not even land plants.

And it's not sea plants either. It goes back to life that got started billions of years ago...and they're still around - quietly giving us the air we breathe.

...they're called "Phytoplankton".

And talk about apologies owed; I guess it's common knowledge where most - if not all - of our most toxic waste is dumped.

Yep..in their home. The oceans.

Bottom line: Tiny ocean plants called phytoplankton contribute 50 to 85 percent of the oxygen in Earth's atmosphere. And they've been doing it before the first land animals - or even the first land plants - existed.

In fact - without their evolution we'd not only not have oxygen - but we wouldn't have the red sediments - iron oxides - that gave us iron and steel....and we wouldn't have Limestone nor marble - both prodicts on planktonic skeletons gathering on the ocean bottoms for ages.

[sciencedirect.com]