The level of copper needed to prevent algae growth even when chlorine levels get to zero is high enough to cause staining of pool surfaces if the pH gets higher towards 7.8+ depending on copper level, though you can get staining at lower pH if you don't manage your copper level and it gets too high. The incompatibility with chlorine is really referring to incompatibility with hypochlorite sources of chlorine (e.g. bleach, chlorinating liquid, Cal-Hypo, lithium hypochlorite) since raising the chlorine level higher as with shocking or even less than daily dosing will raise the pH and that is what can cause staining. If one uses Trichlor tabs, the pH is usually easier to keep lower which can help prevent staining, but the rise in CYA levels over time can still have the water become cloudy from slower and slower oxidation of bather waste (including lotions and oils). Also, copper can turn blond hair greenish.

If you have a vinyl pool, you are at less risk for staining, but if you have a plaster or fiberglass pool, you are taking chances by using copper. Metal staining is a real pain so why take that chance?

Copper is completely unnecessary to prevent algae growth. Algae can be prevented by chlorine alone if one maintains an appropriate Free Chlorine (FC) level relative to the Cyanuric Acid (CYA) level as shown in the Best Guess Swimming Pool Chlorine Chart.