To tell the truth no one really knows why it's possible to grow plants without algae. There is little scientific or empirical evidence that directly applies, and likely not to be any for a long time, as there's just no environmental or economical reason to perform such research.
However, that's a very limited view. Instead view every planted tank hobbyist as a researcher, and every tank as an experiment. Now you have thousands of long-term studies in progress, and though individually they aren't as controlled or in agreement as some would like, together their scope greatly exceeds that of any traditional scientific study.
But it is of course up to you to interpret the results.
To sum up my interpretation:
1) Provide all the plants can actually use, and little more. If there's any question, always err on the side of excess, which is tolerated far better than deficiency.
2) Algae is initially caused by poor plant health or conditions, but once present, it seems to create conditions favorable for itself. Use its presence as a sign that you may not have fulfilled #1, but if you're certain you have and the algae still persists, attack the algae directly.
Agree 110% with this.
I think we can say some about what causes specific species of algae to bloom.
There is strong correlation for some possible reasons, many of which are indirectly linked to poor plant growth, or lack of maintenance and trimming, general care.
Also, if we find one likely cause for algae, say low NO3= BGA.........this does not imply all other possible reasons might not be true, or not, there may be multiple reasons for algae.
Setting up controls for an experiment for algae and/or plant growth implies that the aquarist has complete mastery of light, CO2 and ferts, as well as good horticultural skills, that's a tall order for a newbie to acquire. Even an intermediate hobbyists. They try, but it's very hard for them to isolate and figure out things step wise.
And without a control for the test, they cannot isolate much independently. Some will have good light and CO2 balance, and thus do well with EI, others, not so well, the same can be said for ADA, PPS, PMDD etc, we can find nice algae ridden tanks for any method of dosing.
This is not surprising since algae are not limited in all cases, only the plants are. So there's strong empirical support, but it's falsification, not causation.