Currently most of the ocean is a lifeless desert, with most of life concentrated in the places where upwelling occurs. This kind of floating trees would add enough biomass to compensate for all of the human produced CO2 and even more.
The more refined version of this are intentionally created algal blooms, ala the red tide. Then you would somehow capture the CO2 dissolved in ocean water that the algae concentrates.
In this way you'd use the ocean itself as your carbon capture "filter" and "clean the filter".
The issue is that since there just isn't enough nutrients for life, and adding fertilizers costs more carbon than it sequesters. The deep ocean usually gets its nitrogen from life dying above, decaying downwards then getting pushed upwards again. I don't know of any deep ocean reserves of nitrogen that just needs brought to the surface.
But having real trees above the water is much more useful than simple carbon capture, they would serve as wavebreakers for seasteading settlements too.
There is also emerging technology for nitrogen fixing that could work - Electrocatalytic Nitrogen Reduction.
All this being said, if the goal is carbon sequestration building a viable business is going to be next to impossible.
The easiest way would be to use your renewable energy to pull bicarbonate out of the ocean directly with electrolysis, reverse-osmosis, whatever magic tech the wizards in MIT are cooking up.
Only then you've got the problem of what to do with it all once you've got it on hand.
To bring nutrient rich water up and to generate energy in the process temperature difference between surface and deep water can be used [1]. So if this kind of tree is created, living on open sea can become cheaper and more pleasant than even living on a beach.
[1] https://en.wikipedia.org/wiki/Ocean_thermal_energy_conversio...
Ocean thermal energy conversion - I hadn't heard of this, this is awesome.
Where can I read more about that? My understanding was that it is teaming with microorganisms or near-microorganisms, and of course there are many fauna and flora and protista and other life forms.
For water to team with life, you need materials for building cells in the same place where the light is, and in most cases there is not enough iron on the surface
People really will pretend to believe any old nonsense rather than accept we have to cut fossil fuel use.
you can have that belief, that everyone should give up what oil gives us and live in a pre-modern way that most people would, today, consider poverty, because the alternative is (I assume you believe) extinction.
But you should be honest about it.
So if you think that i like the situation where everyone supports dictatorships like Azerbaijan by buying their oil, you are very wrong.
But what we want and what is going to happen is often very different. Cutting fossil fuel use will not happen until there are better alternatives. And at that point most likely there will be enough CO2 in atmosphere to trigger melting of permafrost which will produce its own CO2. So "these schemes" will be needed one way or another.
That said my main interest here is seasteading, and increasing ocean biomass. Reducing atmospheric CO2 is merely a curios side effect.
Using some genes from mangroves it should be possible to develop a plant that would grow in open sea.