Let me respond again as a private school parent. If you are not willing to have your child go on playdates in homes with nannies and you are not willing to have a child come to your house without a parent, then your child is not going to have many playdates, if any. Once kids are in school, the standard playdate becomes one mom (or nanny) picking up the kids from school and taking them to her house to play. Playdates change from being your friends coming over and bringing their child to play with your child, to being centered on the child's friendships. Even if you get to know the other moms in class and become friends with them, when they ask your child for a playdate, they usually are not interested in entertaining you too. Four year olds should play independently enough that you should be able to handle a playdate plus your other 2 kids. Most moms like playdates because the child with the playdate has built in entertainment, and now you only have to entertain/focus on your other 2 kids.
As for nannies, they are quite common in private school families. If I had not allowed my children to go to houses with nannies, my kids would have missed out on knowing the kids who became their best friends. (We did not have a nanny but I'd guess that at least half of their classes did). My view is that if the family trusts the nanny, then I will too. I grew as attached to some of those nannies as I did the moms. If you quietly volunteer at school and in the classroom, you will hear the scoop on who has the best and worst nannies, and can decide make playdate decisions accordingly. BTW, families with nannies usually love playdates and schedule lots of them.
If other moms at the school (by knowing your husband or by things you have said) are aware of your view on nannies and that you can't handle independent playdates, then it is not so surprising that the PTA moms have treated you as they have and that your child has no playdates. I suspect you are not fitting in with the crowd. That's fine. But if it bothers you, then the private school really might not be the best match for your family.
BTW, my oldest went on his first playdate at age 2. He had started that fall at a private school. About two months into the school year, a mom called me and said the teacher had recommended my child as a good playdate fit for her child. At this point, my child had only had playdates with my friends' children (we all meet at someone's house, we all go to the zoo, etc.). I was shocked that a mom wanted to have my 2 year old at her house without my help to supervise. I had no idea how my child would handle it. The mom sensed my concern and said "oh, don't worry, my other 3 children are having friends over, and it's so much easier with an even number of kids." I'm concerned about 1 kid, and this amazing mom can handle 8. I called the teacher for advice, and she said to let my child try it. My child loved it, and his social calendar quickly filled.
Perhaps the whole play date issue is laziness on my part, but with a two year old having intense temper tantrums, the whole thing just seems overwhelming at the moment. I will however definitely take your suggestion and talk to the teacher. I'm sure she'll have some good suggestions 
And Hannathy, I do not know who the "scholarship kids" are, but as far as I can tell, there aren't any in my child's class. Trust me, I'd much rather befriend those moms and have my son befriend those children than the ones in the PTA.
I bet my kids had a child at our house once for every three times, they went to other houses.
