jbm0148,
Welcome to TFP!
Though phosphates (and nitrates) are nutrients for algae, chlorine alone can kill algae faster than it can grow regardless of the nutrient level. I've had over 3000 ppb phosphates in my pool and prevented algae growth from chlorine alone, but one must have an appropriate Free Chlorine (FC) level relative to the Cyanuric Acid (CYA) level.
Also, enzymes are very useful in high bather load environments as supplemental oxidizers (such as for commercial/public pools), just as ozone and UV and non-chlorine shock (MPS) can be helpful in such environments, but in residential outdoor pools the bather load is so low as for such products to not be necessary.
You can, of course, spend more money on products to do some things faster, but the main principles on this site are to add the minimum necessary and to maintain pools to prevent such problems in the first place. You can take a look at the
Pool School for more info.
Also, with regard to Orenda's products, both phosphate removers and enzyme products, I met Harold Evans who now owns Orenda and we did an experiment on my pool with his products a few months ago in June (he gave me the products free to try). I write about this in
this thread in The Deep End, but the bottom line is that he rarely sees a pool in as good shape as mine (see here and here) and his enzymes didn't react very much in the pool (which was unusual for what he normally sees) and neither the phosphate reduction nor the enzymes resulted in any lowering of chlorine demand or change in water quality (also unusual for what he normally sees).
As for the particular pool in this thread, if the person is impatient with clearing, a polymeric clarifier can be used to speed up clearing. The phosphate remover treatment from Orenda totally clouded up my pool when added (I have a cartridge filter as well) and I needed it clear the next day so I used an old clarifier I had and it worked like a champ. For something like precipitated lanthanum phosphate, chlorine isn't going to get rid of it and it can take quite a while to consolidate in a filter.
Richard