Prevention beats cure on pool algae.
For an existing green pool, you’re going to need something to kill the algae — “harsh chemicals” usually means chlorine, but you can use a non-chlorine shock like potassium monopersulfate or an enzyme-based algaecide. They’re gentler on skin and won’t bleach your swimsuit.
That said, if you have a full-on green swamp, none of the gentle stuff will cut through fast enough. You’ll have to brush, vacuum, and circulate constantly while the non-chlorine shock works. It’s slower than chlorine