From reading comments it seems two blogs are mixed up. One blog has to do with using plants as the (sole) filter in the tank and the other with using high light intensity without using (EI) fertilizers or CO2. I'll start with the planted filter and we can get back to the bright-light topic later if there is interest.
Most plants prefer to take up ammonia rather than nitrate and it takes some energy to reduce nitrate back to ammonia inside the plant tissue. However, I am not really concerned with providing plants with ammonia instead of nitrate. In many cases plants are not energy but CO2 limited so wasting a bit of energy is not an issue. Using ammonia is just a small bonus. In tanks with bacterial filter and dense plant growth I also do not know who will win the fight over ammonia. Does most of the ammonia go straight to plants, leaving the filter with at best a very small population of nitrifying bacteria, or will bacteria convert most of the ammonia to nitrate forcing plants to convert nitrate back to ammonia? In the latter case it may actually be beneficial to discard the filter (or just keep it for mechanical and/or chemical filtration plus circulation). Not so much because plants must waste a bit of energy to use nitrate but because converting ammonia to nitrate is a drain on dissolved oxygen. Darren has written extensively about this and I have started to appreciate this aspect more after finishing the blog (see below).
My main motivation was that I keep and breed wild-caught South American dwarf cichlids and they prefer clean water, i.e. low nitrogen and organic waste concentrations. By giving plants everything they need except nitrogen, the only nitrogen they get is what is produced by fish and decomposition of organic materials. That means their growth rate is limited by how much nitrogen they can access and as long as nitrogen demand exceeds supply no ammonia or its oxidation products should accumulate. By using a substantial, up to 100%, cover of floating plants (that are not CO2 limited and get the highest amount of light) I find I can support fish bio-loads that probably exceed what is recommended for conventionally filtered tanks. Interestingly, I've had most problems with algae after 2-week holiday trips when feeding went from twice daily to once every two days. This reduction of food, and thus nitrogen supply, possibly stunted plant growth giving algae a chance.
People have asked me if they could just remove their filter and trust the plants would be adequate. If you have a well-planted tank with good plant growth and ideally also floating plants the answer is probably yes. But with aquariums I am always in favour of gradual change so processes in the tank have time to adjust. So I would recommend to just reduce the amount of bacterial filter substrate bit by bit during your monthly filter cleaning until nothing is left. If you like the circulation provided by the pump you can keep it going or you can replace it by a small internal circulation pump that takes only a few watts.
Although circulation is generally considered a good thing, I have actually started to use increasingly less flow and several of my tanks have no circulation or airstone at all. This is not because I think less flow is better for plants or general tank health, but because my dwarf cichlids come from the margins of small forest streams and "puddles" where there is limited or no current. I am now searching for very low-flow pumps, just enough to distribute heat from the heater and facilitate oxygen exchange at the water surface. Originally I thought I needed substantial agitation to break up the formation of an oily surface film but now that I have high coverage of floating plants those films no longer form either.
I just had a discussion on an apistogramma facebook group about the wisdom of not having circulation due to its stimulation of oxygenation. My main answer was that my fish show no behavioural signs of needing more oxygen and I have had 10 spawns in the past two months. So apparently it is not a problem. One contributing factor may be that in my tanks there is no oxygen drain by the bacterial filter and I use plain sand, without dirt, in most of my tanks so there is also no decomposing organic matter in the dirt layer. Most of my tanks are also shallow, just 30 cm tall, which gives a higher surface:volume ration. Another factor may be that apistos are relatively tolerant to lower dissolved oxygen levels. I will soon get some pinocchio whiptails that are supposed to require higher oxygenation levels and if observation indicates they need more oxygen I will increase circulation for that tank.