First of all, modesty is important for all parties--for the mom and for everyone else around her. I try to practice modesty, and I am teaching this virtue to my girls. In general, I practice modesty for the benefit of others around me just as much as for my own benefit.
Second, cover-ups are cute and inexpensive. You can find them at any Target, Buy Buy Baby, or Babies R Us. Of course you can find them on Amazon too--like everything else! I have seen several different types of cover-ups and have tried at least two different ones myself, so there are lots of options available. If you don't have an official cover-up, you can easily use a jacket, towel, baby blanket, or even surround yourself with your baby gear such as the car seat and diaper bag. With a cover-up, you can nurse anywhere and hardly anyone will even notice!
Third, you can easily nurse in a private place such as your car or a restroom. I have nursed in many dressing rooms over the years. I found that my babies nurse better when they are in a more quiet and private place. My living room is always full of loud children, so I usually nurse in another room where it is quieter.
Fourth, it is possible to nurse without a cover-up but in a way that will not be noticed by the entire Chick-Fil-A restaurant. Just choose the right spot to sit where you are the least visible and lean forward a little. Find a corner or turn a chair to face the opposite direction. Have your husband sit or stand in a way to provide some shelter from everyone's wandering eyes.
You have the right to nurse your baby anywhere, but life is not all about our rights. Sometimes life is about being considerate of other people. Is it considerate to drink alcohol in front of a recovering alcoholic? Is it considerate to smoke in front of someone who is trying to quit smoking? Is it considerate to tell a mom who has to work that she should be homeschooling her kids?
Do you want to see other people's anatomy? Once your baby is 5 years old, would you want your child to see a grown man scratching himself in public for 5-10 or more minutes? Do you really want your kids to see the plumber's crack? Do you really want your 10-year-old son looking at girls' transparent leggings?!!
Honestly, breastfeeding in public without covering is kind of selfish if you know that it is making everyone around you uncomfortable. It does not mean that you should be kicked out, but don't be surprised if several families decide to leave because they do not want their dads, brothers, or little boys seeing that.
Life is not all about appealing to the masses, but let's be real: if one person can do something small to make ten or fifty or a hundred people feel less awkward, then just do it! For example, if your child is screaming in Olive Garden and is disrupting everyone's dinner, it is not necessary to take them to the bathroom until they quit screaming, but wouldn't it be the nice thing to do?!! Think about how one thing you do or don't do affects everyone around you.