It seems relatively clear to me that just lowering nitrate and phosphate (in any fashion) is not, by itself, typically a good answer to algae problems in the display tank, because if there is an N and P level that restricts algae growth substantially and which also permits corals to thrive optimally, then it is a fine line that is quite hard to maintain. That opinion is based on reading many hundreds of reefers experiences attempting to do so.
If, in fact, folks are more successful in this goal when using an ATS than when using other methods, it may be for one of two reasons, and understanding that reason may well help folks who report not being successful with an ATS.
1. An ats, when properly sized, may be able to maintain that tricky nutrient balance better than some other methods. That tricky balance level may be variable depending on which pest algae are present, but the harder ones to eliminate will be those that seemingly thrive at relatively lower N and P than others (caulerpa racemosa, valonia, bryopsis, etc.).
2. The ATS may be removing something else that algae need, for which corals are a bit better able to attain than algae. I mentioned this earlier in this thread, but i think it bears repeating as most folks assume the effect is from N and P export. If anyone thinks an ATS is restricting pest algae growth when nutrients are otherwise in reasonable ranges (say, 5-10 ppm nitrate and 0.05 to 0.1 ppm phosphate, then it seems highly likely this is the explanation, rather than #1, since many folks have pest algae while in this range.
Further, if #2 is the reason, then folks might fail with ATS if they are dosing whatever that needed material is to levels sufficient for pest algae to still thrive. Iron, manganese, zinc, etc. There's a fairly long list of possibilities of trace element limitations that may restrict growth of algae.