I am a foster parent. When we walked into our pre-placement classes, the first thing the instructor said was that if we were there to adopt a healthy caucasian baby, it would probably never happen. After 5 years caring for foster children, I can say she was very accurate. The fact is that there are a lot of children waiting to be adopted who desperately need a permanent home, but they are anywhere from a few years old to teenagers. Many of these children are very lovable and simply need a forever home, however, they come with a different set of issues than children who have been brought up on your home. In order to get a better understanding of what it is like to be a foster parent and the things you may face, you would need to go to some of the classes and meet some people who are doing it. But, ultimately, if you are only open to a baby straight from the hospital, your chances of getting what you want are very low.
Foster care is extremely challenging, but has been an important part of our lives. Your thinking about a child thinking an older child would think no one else wanted him and the like was somewhat accurate, but even more, the children want to have someone they can trust to care for them...someone who won't give up when things get challenging. They just want to be safe and loved. If you can give this to a child, you are needed.
I hope this has helped. Please contact an agency if you are interested. We work with Focus on Youth in West Chester. They have been great to us and we have been super impressed with their care of the children. We also were with a county near here for quite a while.
I guess the most important thing is to let you know that if you are interested in helping children, foster parents do that in spades.