We're not picking technologies at this point since no-one really knows. Personally, I'm more excited about Direct-air-capture than I am about bio-energy with carbon capture for a bunch of reasons. Storing carbon in oceans is also interesting. Whatever path we see as a truly scalable way (like, can you 10000x your prototype and get costs benefits for example)
I started looking at algae as a way to convert oceanic plastic garbage to something organic.
You can use Molten Salt Oxidation [1] to convert plastic to "synthesis gas" [2] then feed that through a water column to convert CO+H2O to CO2+H2, then recover the hydrogen for power and feed the carbon dioxide to algae tanks.
I was going to try feeding the algae to e.g. eisenia foetida compost worms [3] to make soil and then grow whatever in that. The basic idea was to turn oceanic plastic into biomass.
But then I learned that you can make plastic out of algae. In order to be able to scale the system (without capital) it should be designed to produce as much of its own structure (like a "reprap" self-replicating machine) as possible. The limiting factors should be CPUs and magnets (for motors.)
But this is predicated on being isolated in the middle of the ocean and not having access to capital. I can't evaluate the optimal way of using this stuff in the context of being close to the rest of the economy and with capital, as there are too many variables and I'm kind of ignorant and stupid. (In other words, in a hypothetical universe where someone gave me a big pile of money to try this, the first thing I would do is recuse myself in favor of someone who actually knew what they were doing.)
[1] An exothermic reaction https://en.wikipedia.org/wiki/Molten_salt_oxidation
[2] "fuel gas mixture consisting primarily of hydrogen, carbon monoxide, and very often some carbon dioxide" https://en.wikipedia.org/wiki/Synthesis_gas