For several years, I fed my lawn an organic diet (corn gluten meal in the spring, which serves as a mild pre-emergent, then mixes of cracked corn, alfalfa, soybean meal, often in the form of horse and rabbit feed).
Healthy grass, cut somewhat higher than typical will outcompete most weeds. The only weed I wasn't able to control effectively enough this way were grassy weeds (poa trivialis and poa annua). Triv patches had to be mechanically removed and annua needed late summer chemical pre-emergent (CGM isn't effective enough there). Once those were dramatically reduced (from the "yard" I inherited when we bought the place), the organic system was effective in keeping a >95th percentile lawn. That lawn was more resilient and beautiful than any of the chemically fertilized lawns (whether high or low input) I've maintained. In addition, we had more wild rabbits than I've ever seen, providing additional entertainment for the kids.
The big issue with the organic lawn feeding is that it's quite bulky. No more 15 pound bag of sulfur coated urea that would do two fertilizer applications on my small city lawn. Instead, it was 30-50# bags of organic material that had to bought at Agway or Tractor Supply (necessitating a car trip) and couldn't reasonably be stored for a long time due to bulk and being organic (subject to rotting and rodent infestation). It's also more expensive than the chemical feeding, but still not prohibitive.