Underlying theory of algae control ?

I have been reading these forum articles on algae with some interest. I can't figure out the root cause for algae growth or the reason for why it is controlled when following the general recommendations here. Here are the points that I am seeing. Some help please with pulling the ideas together ?

First, most of what I am reading is focused on optimizing plant growth conditions. For example, that would include EI, N/P ratio, and high CO2. I have not seen Tom say that any one of these, such as the N/P ratio, has a magic effect on reducing algae growth.

Second, the design of EI (as I understand it) ensures that plenty of nutrients are always available. Thus, algae control cannot depend on the plants doing so well that they out-compete the algae for nutrients since we are ensuring that it will not happen.

So at this point I gather that we have created an environment that is great for growing plants and equally good for growing algae. I am not seeing a relationship between growing plants and growing or not growing algae. In other words, it looks to me that one could remove all the plants and the growth or lack of algae growth should not change. Is there something about high CO2, for example, that reduces the rate of algae growth ?

NH3/NH4+ is shown to be a strong promoter. If that is the primary causative for unwanted algae growth, is it possible that a well planted aquarium is controlling algae by sucking the NH3/NH4+ out of solution very quickly ? If this is so, then we should also be paying attention to sources of NH3/NH4+ and our non-plant biological filters. Maybe minimizing the sources of NH3/NH4+ would be the most important and primary factor to get under control if it is not already in control ?