This part of my lawn is still greener and better looking than the rest of my lawn. And it's been two years. Let your dandelions grow. What did a dandelion ever do to you. You can use the young leaves in salads or to make wine.
Children love dandelions and dandelion chains.
Dandelions have deep roots that bring up lots of nutrients from the soil. They add color to your lawn and insects and birds use them for food. I will be honest with you. You may have to change your definition of a perfect lawn when you go organic. Variety is good and a monoculture whether it's a lawn or a farm is not easily maintainable.