My yard is fairly naturalized though I did sprig centipede over most of the open space. But that was as far as it went. I don't fertilize or treat with any kind of chemicals, except in the ongoing battle with fire ants. The centipede is not a thick carpet, but has much open space that allows lots of spring blooming plants to take hold. What most people would consider unkempt, I consider special. We usually delay our first mowing until sometime in May after the covering of May wildflowers of mostly purples, yellows, and pinks have finished their bloom. After the first cutting, I only mow the entire yard about two or three times per season, only maintaining the front yard and main walking areas more regularly. So for us the dandelions just don't work out. When allowed to bloom freely, they will completely dominate the yard displacing most all of the flowers that we enjoy. They also become a difficult item in the annual and perennial beds. So I decided that they had to go. Now we are down to perhaps no more than a few dozen plants, and I'll continue to dig them every time one comes to my attention.
My nearest neighbor, about 1/4 mile away, appears to have dandelion as the dominant plant in the yard. In another week or two, the yard will be completely yellow. There is also that relentless march of the plants on its way up the hill to our yard. Both sides of the road are getting pretty well established. Thankfully, the acreage around our house is all wooded, and acts as a barrier to the dandelions and most other invasive plants.