OP, I'm really sorry this is happening to you. I also don't have a very large social circle of friends (in fact, it's extremely small) And adult friendships are hard!
You've described your friend as being flaky and forgetful, you think that she means well, but she's done this multiple times before. She missed your big birthday, and now you've been trying to get in contact with her about having a meal and she is not answering your messages so, instead of getting ready and going to the restaurant and showing up and waiting for her and possibly either having to wait alone and then finally make that embarrassing decision to leave, or, equally difficult to order and eat alone when you're having a table for two, yeah, I totally hear you. I would hate to be put in that situation.
And yeah, stuff happens. She's her own business owner, this is a busy time of year as a hairdresser because people have parties to go to, and they often need their hair done just immediately before the party, so she might be extremely busy, she has an ex, she has other issues, okay. Adult life is messy and busy and complicated. But, you don't deserve to feel second best or third or fourth. It's not very respectful to your time or your feelings to make plans which then are not followed up on.
I wish I could join you for dinner, I would show up on time. I actually decided to let a friendship of 20+ years go because I felt like I was putting in all the effort, I was making all the plans, I was doing all the kind gestures, I was doing all the favors, I was available when they needed emergency help, I was the shoulder to cry on, and I didn't mind because to me that's all part of being a good friend, but they flaked on me too many times. Where I was left waiting and waiting, or they never got back at all, or plans were changed at the last second, lots of examples, not going to give them here, but I know what you're saying....
Try not communicating with your friend, don't reach out. And see if she reaches out to you. Be mindful that if her message is full of excuses, instead of heartfelt honest regret and apology, if she's being defensive and trying to validate her mistakes instead of immediately acknowledging the hurt she caused you, that's not a good place to start.