So - I'm a newer member, educated by a ton of YouTube pool videos and reading this site a lot, and I'm sure someone more knowledgeable will be along shortly to correct me, but:
From my understanding, there are basically three ways to control algae.
1) Chlorine. If your chlorine levels are high enough the algae will die, including any new spores. It doesn't matter how much 'food' is in the water for algae to feed on because it can't survive anyway.
2) Control food source. This is phosphates. They're constantly being released in your pool by anything that decomposes in it (leaves, flower petals, and notably, dead algae).
3) Use an alternate method of killing algae, like an algaecide.
Issue with #3 is pretty obvious - you don't keep up algaecide levels, so new algae can just grow. You'd have to keep dumping more and more algaecide in each time you see some algae and it will just come back anyway, likely even stronger because dying algae releases phosphates and thus makes it even easier for more to grow. Algaecide often also contains metals and other things that are undesirable in your pool.
Issue with #2 is that you will constantly be getting phosphates in your pool. Dying algae makes it, it settles in from pollen and dust on the wind as that settles in your pool, leaves will get in and start releasing it, etc. You can constantly try to keep phosphate levels low, but it's expensive to keep doing it, and it never stops. And they don't hurt anything to leave in the pool, aside from making it so algae have a way to get a hold if they're able to.
But if you keep up your chlorine levels, algae can't take hold anyway, so it doesn't really matter.
Now, ask yourself which way the pool store makes more money: by selling you (relatively cheap) liquid chlorine, or by constantly having you add more expensive algaecide and phosphate remover?
That will inform the pool company on which method of control they recommend.
