I have also been thinking about this old saying recently.
In past experience, I have seen decent plant growth in green water blooms and even in green dust algae (GDA) blooms. The plants still grew and didn't seem to have any deficiencies, but the algae was definitely all over the tank. Similarly plants still grow perfectly well with cladophora or black beard algae (BBA) growing in the tank in visible quantities.
I think if bloom criteria (trigger) are met, algae will bloom whether or not the plants are doing well. This makes sense with observing algae blooms in the wild. Blooms of certain species of algae appear at distinct times of the year in direct relation to nutrient concentrations, presence of predators, seasonal events (for example: an excess of one nutrient due to rains, or a lack of another due to no rains, etc...).
For example, if a particular species of algae is environmentally cued to bloom when nitrate levels are low for a few days, it will bloom. Since most plants can stockpile nutrients to last them a week at the very least, the plants still have nitrogen available to grow with, they are not deficient yet so the problem isn't that the plant is unhappy which causes the algae to bloom, it is rather the fact that the species of algae is genetically programmed to recognize a particular set of conditions in order to grow.
It is possible that the plant will quickly begin to suffer after an algae bloom gets started since they are already running off reserves and now have to devote more energy and scarce nutrients towards the task of scouring the water for trace concentrations of the lacking nutrient.
I think the statement that "algae = poor plant growth" is usually the case, but I don't think it is 100% accurate in every situation. I think this association tends to be true because people generally don't realize that there is a problem until an algae bloom has been triggered and the plants have depleted their reserves. Since it takes quite a long time for a weakened plant to fully recover and produce heavy new growth after it has been stressed it makes sense that algae, requiring far fewer nutrients to get by, would run rampant until the blooming criteria are fixed and the plants resume their normal roles.
Going along with this idea, I think that the recommended range of nutrient levels (10-20 ppm nitrate, 2 ppm phosphate, etc...) is a range in which few species of algae are programmed to bloom at. I think this may be because over eons of time algae evolution has found it more adventitious to fill a niche in the environment that requires disturbance rather than constancy. For example, if algae and plants grew all the time, provided there were enough nutrients to support growth (i.e. plentiful nutrients 10-20 ppm nitrate, etc...), then they would have to compete with each other constantly for available food which means less energy for reproduction. The plants, being capable of blocking sunlight due to their height advantage and emergent growth could simply shade out any algae that tried to grow constantly causing it to be less successful than a species that was dormant for most of the year except when particular conditions signaled a possible advantage (like a seasonal change for example). In nature it is well documented that species who initially occupy the same niche tend to evolve away from each other over time so that their requirements do not overlap and competition is minimized thereby increasing the reproductive success of both species.
When algae blooming conditions are met (i.e. unstable tank conditions favoring a particular species of algae) the algae blooms until the trigger condition is removed. Sometimes that might mean the algae has found it most effective to simply complete one life cycle and then return to dormancy (GDA), other times it may mean continual blooming conditions until the trigger is removed.
I think we must also recognize that many of the species of algae we deal with in our aquariums have been taken out of their environmental contexts. These algae species have hitched rides from every corner of the globe so particular blooming conditions (like low phosphate levels for green spot algae [GSA]) might be particularly adventitious in the alga's native habitat due to a specific predator, competitor, window of opportunity etc... and now it is not so easy to determine what benefit the algae is getting from blooming in response to the trigger. The relevance of the trigger to the alga's survival has become lost in our home aquarium, but the blooming conditions have not (yet?) changed.
What I would truly find interesting is to find out the original environmental situation of a species of algae that constantly grows in conditions within the recommended range (10-20 ppm nitrate, etc...). I would bet that if such an alga existed that it would occur in water absent of aquatic plants.