For algae to grow they need a full range of nutrients (CO2, N, P, K, Na, Ca, Mg, trace elements, some vitamins etc.), optimum temperature (20-24°C), enough ligh intensity (ideally 50-200 µmol PAR) with long photoperiod of 16:8 (day:night cycle), optimum pH between 7-9, aerating and water circulation + some time to set and develop. If you give them this, then they have an optimum environment for growth ... unless you suppress their growth by some obstracles like algae-eaters, water changes, filtration, low photoperiod etc. Most algae need some time to create suitable environment for their growth (this is called "lag phase") which takes usually 1-2 weeks. But if you do a regular maintenance, clean glass, vacuum substrate, or do a water change, then you actually "reset" them their environment, so that after each maintenance they have to start from scratch. So this all (along with algae-eaters) contributes to algae suppression. So although the algae may have enough nutrients in your tank, other factors effectively prevent them from growing. So you need not only nutrients (especially nitrates and phosphates), but also other optimum conditions for algae to grow and multiply. So it's a mistake to think of only phosphates in regard to algae. But at the same time, it's a mistake to think that high level of nutrients pose no risk.
[BTW, plants and algae prefer NH4 before NO3, so it's better if they have some nitrogen in the form on NH4.]
Also, look at my simple test with 8 kinds of environment and algae growth:
http://www.prirodni-akvarium.cz/en/index.php?id=en_algaeSugars. In distilled and tap water the algae did not develop during 3 weeks.
PS: If you still believe that high nutrient levels don't pose any risk in regard to algae, then try to get rid of all algae-eaters in your tank, increase your light intensity to 100-200 µmol PAR at the substrate, and prolong your photoperiod to at least 16 hours a day (with 8 hours lights off), and add 100 ppm NO3, 10-20 ppm PO4 + other important nutrients into your tank, so that you can be quite sure you have enough available nutrients in water column for algae ... and don't do any maintenance for at least 3-4 week (nor any water changes). Also try to keep your CO2 level on an optimum levels of 10-20 ppm (as higher values may cause inhibition of some kinds of algae). If you'll have no algae after this period, then there may be something on your theory about high nutrients don't contributing to algae infestations.