If you had a big fish load than you would register some nitrates if your test kit was working correctly. I trust API.
It is not just me, just read what Tom Barr has to say about Nitrates as well as Amano. They know a lot more than I or you do with all respect.
Now in my tank, if I let my nitrates crash for even 1 day algae comes back. But not just the nitrates but ALL nutrients if not dosed ball park. PO4 should not crash. I ran other tanks before with similar conditions & trying to keep nitrates low caused massive problems for me. All kinds of algae.
Now if you have such a big fish load why would you not pick up it on your test kit?
I am currently dosing 10-20ppms pf Nitrate an my tank is cranking so much o2 right now that my canisters are blowing o2 out sometimes for minutes straight. Gurgling in o2, & NO ALGAE. Granted I have a light fish load but I am lacking ammonium that you have an I do not. Plants prefer ammonium over nitrates. When I had more fish in my tank the plants grew even faster with the fish + dry Nitrate dosing. At the moment I have a full carpet + heavy plant growth with only medium medium high light . I have to trimat least 1 time a week, with no algae detectable on the plants dosing 10-20ppm nitrate & 1 ppm PO4 with all the other nutrients all being liquid. If you had a lot of algae maybe your filtration is lacking & flow. Also you never mentioned what kind of lighting your running & what spectrum? Presurrized co2 or not?