It feels like there is more to this so I looked back - I'm just going to include this here so others can see the history because it sounds like it may be relevant:
"My daughter had an issue with another student. The teacher's approach (according to her it was her own decision) was to not allow the girls to hang out or speak to each other. Soon, my daughter was being excluded by the other girls. Parents were choosing sides. And my daughter was losing friends. I asked the principal for such "rule" to be removed, & asked for another solution. To keep it short, the other girl's parents moved her to another school. "
I'm glad things seem to be getting back to normal but I can see why this year has been difficult for your daughter and no doubt you. You said the teacher has been awkward with you, and no doubt the parents have been a bit stand-offish.
For the teacher to bring up that your daughter is having trouble connecting with kids this year sounds understandable, given the circumstances.
I would be more concerned if she didn't get along with children - period. You added that she has playdates with another little girl and they get along well. To me, that's a very good sign. My daughter's two good friends are not classmates. They were made through activities. One of my kids had mostly neighbor friendships until later - more like 4th grade when he then would go over to friends' homes from school. He was old enough to arrange it himself and it then took off.
As for not having a bestie at school - only one of mine did at that age. I think only one actually even had a loose 'group' at that age. My kids kind of just hung out with whoever was playing whatever they were interested in that day. Some of mine are introverted and quirky. They can be shy also. The not greeting each other out in public thing - I know that. You must have noticed that with other children too though at that age - I doubt it is just your daughter. Just keep reinforcing the behavior you want to see.
I just finished reading Queen Bees and Wannabees - it's a book recommended on this site frequently. It was interesting. One thing it says frequently throughout the book, and why I mention it, is to check your baggage! You mention how you feel about these moms - that it reminds you of high school - don't let that affect how you view your daughter's situation at school. The two are separate.
Quite frankly, you're not responsible for your daughter's friendships. One of the moms wrote that on here once, and it was the best advice. We can help our children learn social skills, and model appropriate behavior, but your friendships should not be the basis of your daughter's friendships by third grade.
I think you might benefit from seeing a counselor. I did this and they just got me seeing things from a healthier perspective. Not feeling like you're screwing up your kid is very liberating and just means you can focus on being supportive and empowering, rather than thinking 'What is wrong with my kid'.
If you do find though that your daughter is struggling, seeing a counselor hasn't helped her (she can always benefit from a few sessions also), and this is all just too much and a fresh start would just be the best thing - then maybe a new school would be a great idea. My kids changed schools and did well (we moved).
Keep in mind - if you move, the other kids will still all have known each other and there will be established friendships so that will still be a challenge regardless of which new school she goes to.
Best to you and your daughter. Keep us posted :)