Algae grow when there is a great chance that they can complete a growth cycle and reproduce. All other forms of life also expend most of their energy trying to reproduce. Aquariums have been around for perhaps 100 years, but natural water bodies have existed for something greater than 1,000,000 years. It should be obvious that algae evolved to be able to reproduce in natural settings, not in aquariums.
It appears that in natural settings algae and plants compete almost exclusively for light. Naturally growing aquatic plants tend to grow to the surface and cover it to get as much light as possible, while algae have to stay where they are, under the water surface. (Yes, this is a generalization.) In order for algae to have the best chance to complete a life cycle and reproduce they have to start growing when there are few aquatic plants growing, when aquatic plants won't be covering the water surface until after the algae have completed their reproduction. So, algae evolved to sense things that indicate that they will be successful in reproducing.
Two of those things seem to be the sudden presence of ammonia (dying plants and other water life?), and a drop in dissolved CO2 in the water (?). Once those things tell algae to start growing they will grow tenaciously until they finish what they are trying to do - reproduce. It takes very little nutrients for algae to grow, compared to plants. The consumption of nutrients is roughly proportional to the mass of plant or algae tissue. Algae, even when abundant, has very little mass, but plants have orders of magnitude more mass. So, algae don't need nearly as much of any nutrient as plants do.
Algae does not compete with plants for nutrients, because they don't need to. There are always more than enough nutrients for algae in our tanks. (Keep in mind that we are not discussing natural bodies of water now.) These things mean we should try to avoid any of the "signals" that tell algae to start growing. We know that ammonia will trigger algae to start, and we know that allowing the concentration of CO2 in the water to drop will also trigger algae to start growing. We also know that plants consume ammonia very quickly when it does appear. So, if we have a lot of fast growing plants growing, ammonia never exists in the tank except in very tiny amounts for very brief moments. That is why having a lot of fast growing plants, which means having adequate nutrients for those plants, is a protection against algae.
Is all of the above really true? Who knows? It is certainly an explanation that fits the known observations by a lot of people. So, it has a good probability of being true.
Natural bodies of water are another story entirely. Not all of those bodies of water have any aquatic plants in them. So, what triggers massive algae blooms there would likely be different. Other natural bodies of water have very little CO2 dissolved in the water - again algae blooms would be different. But, some natural bodies of water have both high CO2 content and lots of healthy plant growth - those should behave similarly to our aquariums.
There are at least thousands of different algae on earth. We face only a tiny percentage of those in our aquariums. What happens in nature may well involve algae types we never see in our tanks.