Not all Algae is photoautotrophic
The purpose of an algae turf scrubber is to grow and harvest photoautotrophic "turf" algae, such as derbesia.
CO2 can be produce by bacteria or yeast that break down organic carbon in the aquarium.
Since microalgae have this ability, their carbon metabolism can be divided into four types (1) photoautotrophy, (2) heterotrophy, (3) photoheterotrophy and (4) mixotrophy.
source of carbon for cyanobacteria growth can be organic substances, such as sugars, fatty acids and amino acids. However, the ability to grow heterotrophically or mixotrophically on one or other of the organic substrates is species dependent
You definitely make an interesting point. There is a carbon cycle in the system. Regarding heterotrophy vs autotrophy though, I'd argue though that really, other than some species of dinofallegellates, I don't believe that any "algae", common to our systems are in any way heterotrophic. And certainly the purpose of an ATS is not to harvest heterotropic organism, but rather autotrophic "algae". Also, as you mentioned, Cyanobacteria can be mixotrophic, but is considered to be bacteria. Both cyano and dinos are generally considered nuisance organisms, if growing on an ATS anyways (perhaps we should harvest cyano, but that's another discussion), but regardless, are not generally (I say generally as I know of at least one species of cyano which, IIRC, was shown to grow as well in dark "heterotrophic mode" and light "autotrophic mode"- this appears to be an exception not the rule though) as efficient when living in heterotrophic modes. Some species can use heterotrophy for survival, but it isn't usually their primary "mode." So, here, I'll refer to algae as autotrophic and bacteria as heterotrophic (as, it is essentially is, at least for all reasonable intents and purposes as far as properly functioning ATSs are concerned).
As far as removal of organic carbon, as mentioned, bacteria can use DOC, and produce CO2. On the other hand, algae can consume CO2 and provide some DOC to the system (such as if there is excess after energy requirements are met). So, bacteria can benefit from the algae. Basically, the bacteria will be limited by the carrying capacity of the system and if this capacity is due to a carbon limitation, then the algae could provide more DOC for the bacteria, thereby, increasing the carrying capacity for bacteria. However, nutrients (N, P, Fe etc..) used by the bacteria will not be exported from the system unless the bacteria is removed from the system (and likewise if algae is the sink, will not be removed unless the algae is removed). At the same time however, CO2 is a noble gas and follows the ideal gas law. So, the CO2 level is dictated by the partial pressure of CO2 (pCO2). Therefore, assuming there is reasonable gas exchange, there is really no reliance of the algae on the bacteria, for CO2. For practical purposes, the algae is being removed, so, consumption is limited, and in comparison, the CO2 sink, when talking about a small system, is therefore essentially unlimited. So, the alga isn’t dependent on the bacteria.
However, the carrying capacity of algae and bacteria is, in both cases, also limited by the nutrients in the system (iron, nitrate, phosphate). The autotrophic algae however, as mentioned, aren’t reliant on the bacteria for energy, but are still competing with the bacteria for these non-carbon nutrients. The bacteria are reliant on organic carbon though, which needs to either come from food input, or the algae. If nutrients go to the bacteria, and your not exporting the bacteria, and these nutrients are not exported. If the ATS is set up correctly, your ideally creating an environment that shifts the competitive advantage for these last nutrients to the autotrophic algae. You harvest the algae, so, there is no space limitation and the algae can continue to grow and consume these nutrients. The carrying capacity for the algae is not met. So, if the size of the ATS is matched properly to the bioload of the system, nutrients decrease as the algae grows and is removed. However, still the net organic carbon does not decrease. Carbon used for metabolism is cycled, but carbon for growth remains in the algae or bacteria until that organism is removed. The nutrients coming into the system also bring carbon roughly in a standard proportion (such as the Redfield ratio). So, carbon for growth is basically fixed, but CO2 allows for an extra input. So, if the competitive advantage for resources is algae dominate, the carbon at best neutral. In order for organic carbon to decrease, there needs to be some form of mechanical filtration.