Thanks for the observations and experiment! Here are some things I gleaned from the OP and various responses. We've got similar backgrounds, but my focus ended up in biogeochemistry and aquatic plant ecology more than the straight out water resources side. Please don't take my reply the wrong way, I'm restating some things for the less scientifically inclined readers.
1. Goldfish are really messy.
2. Stem plants can trap detritus.
3. Anaerobic bacteria are good to have to reduce NO3.
4. Methanol can effectively help lower NO3 levels
1. Yes, goldfish are super messy and require a commensurate amount of maintenance. They're the crap factories of the fish world, no doubt about it.
2. Yes, thick groups of stems can and do trap detritus. That being said, careful siphoning will suck it out. Using a hose without the large siphon tube will do the trick. If not, swish the debris out with your hand or a powerhead and then siphon it out. Those have all worked for me in the past.
3. Aerobic, not anaerobic bacteria benefit most from organic carbon supplementation. Aerobic metabolism is much more efficient than anaerobic metabolism so the buggers will take up more NO3 and PO4 than their anaerobic cousins. On the flip side, that doesn't permanently remove NO3 and PO4 from the system like denitrification does. It's just a temporary sink, however long-term, as the buggers assimilate N and P into their bodies.
4aa. Yes, NO-POx is a safe and effective means of adding dissolved organic carbon (DOC) to aquariums. Some folks may not think it's worth the price and choose to use other materials, but the product itself is safe and reliable; moreso than non-denatured alcohols and potentially vinegar.
4a. The supplementation of organic carbon will stimulate bacterial proliferation only when DOC is limiting. This is particularly true in reef systems with a good protein skimmer and/or limited feeding. With all of that fish waste in your Goldie tank I'm skeptical that there's a DOC limitation, but that doesn't mean there isn't. The flip side of the coin could be that there's that much NO3 because there's a lack of DOC and for the bacteria to effectively take up the NO3. Again, I'm open minded but skeptical as Bill Nye likes to say. The same could be said of PO4. DOC, NO3, and PO4 are all legs on the three-legged stool. When they're in balance bacteria do well. When one's limiting then the other two legs get longer and the whole stool tips over.
4b. Yes, anaerobiosis does reduce NO3 levels. That's why people used to use denitrators on reef tanks and is how Seachem's Matrix accomplishes the same. As Wastewater stated, this is also quite effective in large scale water treatment.
To compare aquariums to wetlands is a bit of a stretch as there are seasonal effects, water level fluctuations, and nutrient input variables that make wetlands the important sink they are. In an aquarium environment things are a bit different. Firstly, there's no drainage or significant subsurface water movement to transport nutrients like there are in wetlands. Stimulating anaerobic zones can quickly cause problems in a planted tank. This is different than a reef system where the rock is full of small holes and crevices that get good water circulation/nutrient input and become anaerobic in their depths. I'm a firm believer that this is the main source of denitrification in most reef tanks as the typical substrate material is far too small-grained to allow sufficient diffusion of NO3 to be a significant sink. Polychaete worms and other burrowing critters may add enough crap to keep it going, but that's not NO3 coming from the water column.
Another point going along with the above; in order to distribute the DOC/alcohol to the bacteria in a planted tank
in the first place there would have to be enough water movement to the area to accomplish this, and that means concurrent distribution of O2. Therefore, at best (as far as denitrification is concerned) you'll have the facultative buggers getting short bursts of growth and NO3 uptake then going back to their usual slow anaerobic metabolism. The previous statement discounts possible preferential flow paths around plant roots that may distribute the DOC to hypoxic or anoxic areas in the rooting zone. There again though, you have a source O2 and DOC exuded from roots so you've got a source of both for at least a portion of the day.
5. Heavily stocked tanks will have higher levels of waste byproducts. This is especially true of the fish are crap factories.
6. The best way to lower NO3 and PO4 in a planted tank is robust plant growth and removal of waste/detritus. If levels are too high for your liking many of the larger floating plants are excellent sponges for these, grow quickly, and are easily removed. Plus, your Goldies might like eating them too. If that doesn't work well enough for your liking or you can't stand the floaters then perhaps DOC addition is the right method for you.
I'll be happy to go into more detail on specifics if anyone has questions. Constructive and respectful debate is good too.