Chloroplasts should not be able to work once they are removed from algal cells because those cells contain proteins that control their operation.
You might remember seeing single-cell organisms under a microscope in biology class, and perhaps you remember them as rather boring blobs. Life got much more interesting and complex when the cells joined together to become animals or plants. Most people, biologists as well as you and I, think of cells primarily as building blocks of more complicated organisms. Learn more on this Moment of Science.