You (and your daughter) are working towards two very different goals: being a confident well-developed person, and, having a teenager best friend. It sounds like your daughter is effectively portraying an image of being "her own best friend" - not a bad way to be in life, but, not the best way to attract another teenager to be her close friend. Your daughter sounds like she portrays an image of someone who does not "need" (which can look to other teens like "does not want") anyone's friendship.
Is there any girl that your daughter really makes an effort towards, really shows that she cares about the one person?
To "switch around every day" the group she spends lunch or recess with - how could that possibly help her to develop a close friendship with any one person at school?! It's a great technique for exposure to many types of people (as I said above, being a "well-developed person"), but NOT a good technique to make any one schoolmate feel like your daughter really cares about a friendship with them personally.
And with all of her teams and activities in other towns, it sounds like your daughter is just flitting from place to place, from social group to social group, all the time.
Your daughter sounds very accomplished! But part of *having* a close friend is *acting like* a close friend. Wanting a friend when it's convenient for her (for example, the fun of a three-person costume because your daughter will be in the neighborhood that night so she wants to "line up a friendship" for that night), is not the way to get that true close friend.
Encourage your daughter to "pursue" what she wants. If your daughter really wants a few close friends, she should first identify the best "candidates" and then she should focus her efforts on really spending time with those people and having good conversations with them, not just floating by them on her way to another lunch group or another activity.
(ETA: In your SWH you say she is "very happy" but in your original post you say she is "very sad"!)