I run an ATS as part of my nutrient export system. I also run a large skimmer, GFO, GAC, polyfilters and have a large chaeto refugium. Hard for me to say, therefore, how beneficial the ATS is other than my 'beleiving' it's an important component of the system. My logic is thus:
1. Skimmer removes nominally 30% of the organics before they break down.
2. Of the remaining 70%, some are removed by the GAC, but the majority degrade to nitrates, phosphates, etc.
3. Since my tank is more likely to be nitrate limited than phosphate limited, I assume the GFO removes much of the latter.
4. ATS/chaeto (both grow well) then removes the nitrates and remaining phosphates to the point they are undetectable.
5. GAC removes any greening from the ATS; polyfilter removes any heavy metals that might find their way into the system.
That's my story, and I'm sticking to it!
FWIW, I have 45 fish in a 265, feed a LOT, and have no measurable nutrients. I'm actually experimenting with a DiY chaeto reactor.