No, NO3 stays around0.2ppm to 1ppm without it in my tanks. Zoanthids are doing very well. I do dose vodka and vinegar to keep nitrate and PO4 low; perhaps the bacteria are helping as part of the food chain. PO4 runs 0.02ppm to 0.04ppm per hanah 713.
Sorry about your zaontids. Any predators or pests? Lystmata wundermani( peppermints), asternia stars eta lia? Sometimes , an out of tank bath in 4parts tank water and 1 part 3% hydrogen peroxide for 5 minutes can perk up a colony and remove pests as well as many types of nuisance algae. Zoanthids tolerate that treatment well.IME. BTW, blowing accumulated detritus off and away from them seems helpful.
You might try a tiny bit of sodium nitrate if you think nitrogen is deficient.
What's the salinity.btw?