If you measure the nutrient levels now when the pond is green you will probably find the levels are very low, maybe even zero. This will be more true if it gets greener and greener. A pond that has been green for awhile will virtually always measure zero nutrients, or at least zero in one key major nutrient.
Should the pond then suddenly clear, as they often do, you can measure the nutrients again and you will find that the nutrient level increases, a lot. As the algae dies and decompose they release the nutrients they consumed earlier, plus you still have the pond/fish producing lots more nutrients daily.
When water clears naturally you can normally add handfuls of fertilizer and the water will stay clear.
The concept that plants remove enough nutrients to kill algae is very old. Something repeated thousands of times. But so easy to disprove.
After a pond clears the next question people often have is "how do I control string algae". The plant starving algae thing seems to only affect single cell algae for some strange reason. One of the more silly myths.