Nitrate in the tap (source) water is one issue, and nitrates occurring within the aquarium from the biological processes is another.
Nitrate of 20 ppm in the tap water is not likely to lessen in the aquarium, so right off this is what you have to work with. There are means of dealing with tap water nitrates, you can have a read of the articles linked below from one of our members.
To the nitrates occurring within the aquarium, these can be reduced by regular substantial partial water changes, vacuuming the substrate in open areas, not overstocking, not overfeeding, and having fast growing aquarium plants. Some of these you mention doing, so that's good, but the plant aspect is often misunderstood. Plants need nitrogen, and the majority of aquatic plants we keep in an aquarium use ammonia/ammonium, and they can take up a lot of it. Unlike relying on nitrifying bacteria, nitrites are not produced, which then means less nitrates. Some will still occur, but far less than having no aquatic plants. Fast growing plants are best at this.
To your Hornwort...this is not always an easy plant to grow well. You would be better with more substantial floating plants like Water Sprite, Frogbit, Water Lettuce, or stem plants like pennywort. Floating plants are right under the light so this usually means easier to grow.