The problem is that if algae dies, it's most likely die at the same time making a sudden and great O2 shortage making animals die, which creates the same process.

It's more about the imbalance caused by algae blooms. They breed prolifically, and die off en masse more or less constantly as they bloom. When they die, they decompose and release carbon dioxide back into the water. So algae blooms hoover up carbon dioxide and concentrate it in a specific spot of ocean water, which can cause problems regarding anoxia and also ocean acidication.

The issue is that after a couple hundred years of intentionally eating literally everything in the ocean and dumping tons of our garbage and industrial waste there, oceanic ecosystems are even more fragile than usual and we don't exactly have the ecological spare room to tinker with wild algae blooms on a scale large enough to make an impact on climate change. It would be trivial to ruin oceanic ecosystems, and by extension, many land-based ecosystems, with a megascale algae bloom.

Vats of algae in controlled environments might be a way to go, though?