In my experience the production of nitrates can be naturally achieved if the correct conditions are available to produce excess nitrate levels. In my first saltwater tank I struggled with high nitrates 80+ppm for years until I finally understood that my methods of filtration were causing the problem. My high nitrate issues were related to having canister filters and quick sand filters. The quick sand filters are from the 1990s and are similar to a fluidized bed filter. Both of these types of filters were nitrate factories as they were very efficient at breaking down waste to ammonia->nitrite->nitrate but they were not efficient in breaking down nitrates to N2, N2O, or other nitrogen compounds. Filters that trap waste, have a large surface area, and lots of oxygen available have a higher likely hood of producing excess nitrates. The resulting imbalance in the nitrogen cycle allows excess nitrates to build up because of the lack of adequate denitrification to keep nitrates balanced. A large portion of denitrification takes place in anaerobic conditions, where the bacteria that consume nitrates thrive. When I finally defeated my nitrate issues and got the levels under control I made a denitrification filter that provided surfaces area for bacteria in an anaerobic which balanced out the nitrogen cycle in my system. Since you are faced with the opposite problem and you would like to increase nitrates you can try to create an imbalance in the nitrogen cycle that produces excess nitrates. Some methods that might work for this would be using filter socks or filter pads and just leave them in place without being changed or cleaned. Add a canister filter with filter pads and/or biological media and neglect the cleaning filter. As with any change to a system I would caution to take it slow, observe your tank inhabitants, and closely monitor water parameters.
Regarding the utilization of nitrates by corals I am no expert in that process nor have I read and of the documentation cited in this thread. I would like to add to this conversation though as I find it to be an interesting topic. One thing I have learned about reef aquariums throughout the years is that the number of variables in the mix at any one time is vast. Considering the vast number of variables it can be difficult to understand why a change in nitrate levels even if its minor creates an observation of increased coral health or growth. While nitrates might not be consumed directly by the corals there is a potential that the increased nitrate level impacts other life in your aquarium which might benefit the corals. As a theoretical example lets say that a minor increase in nitrate levels allows the natural phyoplankton and micro algae population to increase, which in turn creates a larger food supply for other mico and macro fauna such as copepods which as a result of a larger food supply increase in population. Increased populations of micro and macro fauna means that the number of organisms breeding will be increased due to the higher population. As these small organisms breed they will release eggs and sperm into the water or in some cases larval stage organisms or small adults will also be free swimming in the water column. All of this small life has a chance to to captured and consumed by corals as food which can benefit the corals. Again that was just a theoretical example but it can't be ruled out that increased nitrate levels can impact other organisms in an aquarium system which can directly benefit coral health.
