To continue: "Stabilized Chlorine" makes you think it's "different" kind of chlorine from bleach. It's not. Chlorine is chlorine. But as Tri-chlor dissolves, it releases Cyanuric/Isocyanuric Acid (aka, CYA), which is chlorine stabilizer. You can get the same effect by adding bleach and adding granulated stabilizer/CYA. Tri-chlor is also very acidic and can lower your pH quickly.

While algae IS harmless (as kelemvor points out), it is the canary in the coal mine that your chlorine level is too low to kill the harmful stuff. Further, your chlorine is spent killing the algae and isn't available for the bad stuff. So killing algae is necessary to keep enough chlorine to kill bacteria and viruses.