T.F.
I believe your expectations are too high for a 4 year old. She is 4, not 14!!
My daughter, now 21, is my only child. She was raised around other "like" families from day one, went to Mother's Day Out, Preschool and then started her way through the educational system. We went with public education because our area has one of the best systems in the country which I pay high dollar taxes for and it was worth every penny.
In the beginning, I happened to have 3 neighbors who were friends and we all had our first baby within 2-4 months of each other. We got together daily with our children and even though we all moved to different school districts within the Plano/Dallas area, we remain friends to this day and so do our children.
I think it is important to have friends from multiple outlets... as in not all from school, preschool, etc but look into whatever sport/activity your child is involved in, your church, other activities.
My daughter had a certain group of friends from her martial arts family, orchestra, cheerleading and so on. These friends change from time to time and that is normal.
I've been in the school system teaching as well about 15 years and children develop friendships at different levels. I see A LOT of situations with the best and dearest friends being those whose families were close and socialized together while the children were growing up. Prime example is the neighbors we had a bond with from day 1 and that bond will always be there with those families and friends, no matter where we end up living.
Your daughter is 4. Please don't let your high expectations and anxiety about friendships rub off on her. Believe me, she picks up on your anxiety. YOU model the behavior YOU want her to have. Go to the library story times, book store story times, etc. There are a lot of opportunities to meet other parents and children that cost no money is that is an issue.
Be patient and give her time... she will be 21 before you know it!! Enjoy the ride...ups and downs and you will both grow.