This is an old question, and I'm not answering many things this year, but I'm not sure this question has been answered clearly before, so . . .

Chlorine -- whether bleach, trichlor, dichlor, cal hypo, or from a salt water chlorine generator -- is added to pools for THREE reasons:
1. To sanitize the pool, by killing bacteria, viruses or -- eventually -- even amoebic cysts, like the agents that cause cryptosporidiosis (https://www.cdc.gov/parasites/crypto/index.html)
2. To oxidize, or chemically burn up, people 'goo' including pee, poop, sweat, snot . . . and various lotions, oils and moisturizers.
3. To kill, or prevent, algae.

Hydrogen peroxide IS used in pools, in the Baquacil / Soft Swim / PHMB pool treatment system. Under this systems, THREE chemicals are needed to replace just ONE --chlorine:
1. PHMB sanitizes; and
2. Quaternary ammonia kills algae, and
3. Hydrogen peroxide oxidizes.

It should be clear that hydrogen peroxide can NOT replace chlorine because it only does 1/3 of the work that chlorine does:
1. It does NOT sanitize (at levels used in pools);
2. It does NOT kill algae (at pretty much any level);
3. It does oxidize 'people goo', but NOT as well as chlorine does.

It is possible to have non-chlorine systems that use peroxide. I actually installed one, many years ago, on a VERY expensive indoor pool. But this system used UV (for sanitation) and polyquat (for algae) and peroxide for oxidation. It *WAS* very, very nice to swim in -- I swam laps in that pool several times.

But, the system cost over $10,000 over a decade ago AND it was high maintenance . That was not a problem for the owner, who was not a billionaire, but didn't miss it much. But most people, even well-to-do people, will be unhappy with a system that costs $20,000 today AND requires monthly service calls from a VERY highly trained (= VERY rare) pool guy.

What's more, this system would NOT have worked out doors, the way I built it. I do know how to make it work outdoors, today, but it would still be a very expensive system. And, there are probably less than a dozen pool guys around the country who could design and build it, and then service it.

Bottom line?

Chlorine is NOT always the best pool chemical system. But it almost always the best PRACTICAL pool chemical system.