Algae are interesting, but I don't see them as being ideal for tying up the carbon for long - we'd need to compress it into bricks and bury them, perhaps?

Aren't we already growing algae in the ocean about as fast as we can? Is there more we could be doing than manufacturing fertilizer on a geological scale and letting a lot of it end up washing out to sea? (Honest question, I haven't thought about this avenue of geoengineering before)

It's not immediately clear to me that the energy involved in creating a "more controlled environment" would make the investment a net positive.

In a contained setting it still will sequester carbon. It is more easily monitored for expected behavior. It minimizes risk to other ecosystems.