My experience disagrees, but with an "It depends".
Case 1.
In a situation with high particulate food (live or otherwise) that is not simply settling and decomposing in the algae, the coral will be able to utilize that, the algae will not. Coral wins as free floating nitrate is removed. This is the situation on a real reef, and perhaps in an established tank with good diversity as well.
In the case, we talk about TOM ( total organic matter) and not about nitrate but nitrogen, TOM of which the nitrogen content for +- 85% will be turned into ammonia-nitrogen after digestion ( remineralisation and dissimilation), to be used by algae and other organisms. Wich means from total TOM removed a lot more nitrogen becomes available for algae and other growth as used up by the coral.
Case 2.
A situation without those non-free nitrogen sources of food. If the coral is relying on free nitrate (ie not bound into a particle or organism) then the "level to sustain coral" is > than the "level to sustain algae". Coral loses. Algae wins.
Corals have their own holobiont, taking up inorganics to produce protein, providing organic nutrients to the coral. The Symbiodinium may provide the energy source. Why algae should win? A coral with symbiodinium does not need to be fed organics to fullfill its need for organic nutrients. The coral may manage its nutrient suply by managing the growth rates in its own holobiont by providing an organic carbon source and other nutrients by releasing mucus. The carbon is fixed by symbiodinium into suggars, used for energy production. Corals do not like an environment with high DOC availability as they may not be able to mange the holobiont growth rates as desired. in a normal situation DOC is used up fast as it is limited available because most of the carbon is released as CO2. I think corals have nothing to fear in the competion for nutrients with algae using nitrate as a nitrogen source.
Free available nitrogen is ammonia-nitrogen already present in the water column which is toxic. The production of ammonia-nitrogen from TOM is not free at all as a lot of energy was needed to produce DOC, for respiration, dissimilation, the production of CO2, phosphate and ammonia.
Almost all microorganisms prefer ammonia-nitrogen as a nitrogen source for growth, most inhibit the use of nitrate-nitrogen if ammonia-nitrogen is available. A lot of algae are able to use both ammonia and nitrate if ammonia is limited available and are able to use both pathways at the same time. If using nitrate growth rates are slowed down and is often used only to support the cells, waiting for ammonia-nitrogen to induce growth. As most stony corals are supported by is symbiodinium and holobiont for there energy source and nutrient supply they do not depend much of what you call "the level to sustain". My opinion, there is no such level, only when there is not enough or in the case of to much DOC . But something must limit growth and in most cases, it is the growth rates creating unavailability. High growth rates are linked to the availability of organic carbon and the use of ammonia as a nitrogen source. It is not the level of nitrate present in the water column which will be responsible for the growth rates wich may lead to phosphorus starvation, it will be the demand on that place where the action and competition take place. Biofilms, microbial mats, coral holobionts, etc.. are able to manage there own food supply for a long period of time without what you call " non free nitrogen sources of food". A coral holobiont loses that ability in an environment with high DOC availability. This means DOC management seems to be a lot more important compared to nitrate management as nitrate does not harm a thing, except at very high levels.
The nutrient level does NOT influence growth rates, it is availability. The use of nitrate slows down the growth of algae. The coral is supplied ammonia by its holobiont. Why algae may win using nitrate as a nitrogen source?
Case 3.
No particulate food, no free nitrate. Cyano bacteria can fix their own nitrogen and will win. Everything else will eventually starve or get over-run by the cyano.
There is no such thing as free nitrate. Nitrate is the end product of aerobic remineralization.
To produce nitrogen usable for growth a lot of energy is needed. Transforming nitrate into for growth usable nitrogen needs a lot more energy. Fixing nitrogen even more. Also cyano prefere ammonium-nitrogen and will inhibit nitrogen fixation if other nitrogen pathways are available.
Food starved organisms provide nutrients for others. What is the growth-limiting factor? I prefer it to be nitrogen if I had the choice.
Organisms grow at a logarithmic rate and die a natural death at a logarithmic rate.
Not all cyano can fix nitrogen and also some other organisms have the ability to fix nitrogen. They only fix nitrogen if there is no other nitrogen source available as fixing nitrogen is very expensive needing a lot of energy. But they have an unlimited carbon and energy source not depending on availability in the water.
In microbial mats, the nitrogen source is provided by natural remineralisation within the biofilms. In these conditions, cyano may support the microbial mat in a nitrogen limited environment. Cyano and other nitrogen-fixing organisms are very important members of the coral holobiont.
In the ocean, they play an essential role in the nitrogen cycle and as a food source, as a member of phytoplankton. Cyano is very effective in storing phosphorus and most of them are able to fulfill 5 cycles based on their phosphorus reserve. One does not want both, phosphorus- and nitrogen-limited conditions in a reef aquarium.
Case 4.
Case 3, or pretty close to it, but with a phosphate shortage as well. Dinos might win. Coral and other algae still loses.
Phosphorus unavailability is a killer on short notice. That is why most organisms are able to store a phosphorus reserve. Bacteria known as PAO contain a very high phosphorus reserve which they may release in nitrogen-limited conditions. Symbiodinium are dino's!
Nobody will win from phosphorus shortage but PAO ( phosphate accumulating organisms) , including cyano, will support microbial communities wich may recycle phosphorus, enough to support the comunity for a period of time. Dying microbial communities are the messenger for a lot worse. Algae may loose +50% of their weight for supporting the comunity. Will dino's win? As members of a microbial comunity or member of phytoplankton?
Algae vs. bacteria winning in other cases is dependent on carbon availability and other things, not just nitrate levels.
It has been shown nutrient levels do NOT influence growth rates. It is availability. Total growth of coarce will be influenced by total availability. The availability of DOC, and the available nitrogen source used will be determent for wich organisms will be dominant at that moment. Heterotrophic bacteria, r-stratigists, will always out-compete other organisms for ammonia-nitrogen in the water column if usable DOC is available, certainly when carbohydrates are added. When using nitrate most photo-autotrophic organisms, having a free energy source, will be very competitive for the available sources. Heterotrops will use up natural available DOC anyway. Other autotrophs, AOB, NOB, AOA, will be outcompeted if not in a protective environment. Adding organic carbon will replace the autotrophs by heterotrophs.
