I've had more success writing Self-Inserts than anyone else I know, so take it from someone who knows. Rowling had FOUR self-inserts in the Harry Potter series.
First is Hermione, the publicly admitted one. She was supposed to be the 'image of childhood past'. But you'll note a lack of author-empathy with the character later on. This is a tell that, though she may still have a hand in the sock-puppet, this *isn't* the character Rowling truly sees herself as. Not now, not anymore, at any rate.
Second is the next most obvious one, Ginny is the 'image of romantic self'. Rowling feels no need to describe this character, or why she would be desirable to Harry, because Rowling inherently believes she *is* desirable, and so her character is also, and naturally she, being in charge, gets the boy.
Both of those and their relationships could be described in a great deal more detail, but I won't bother. It would all be rehash anyway.
Third most obvious is, naturally, Dumbledore. He is her 'author-insert' for lack of a better term. He is her plot device, her avatar, her all, in a 'author walking the pages of her own story' sense. He is the creation she uses to get everything done. Rowling doesn't have to think of reasons. Anything she wants done, Dumbledore causes to happen. He is her fingers in the story, and the reason he comes across as a monster later on is basically that Rowling never thought through any of her actions, so nothing he does adds up.
This was not a character built with driving reasons, a personality, background or goals. This was her lazy man's way of getting done the things she wanted to happen, without bothering to think up how they might have turned out that way on their own. Feel like writing a story where the main character is abused? She never bothers to ask herself how that could have come to pass, she just makes Dumbledore send him there. Then back again, etc. Want a tournament in which people die? Dumbledore organizes it. Want a secret organization to fight evil Voldy? Well, of course Dumbledore is in charge of it.
If you take a bunch of random story ideas and drop them all in a bucket, mix well, and then use Dumbledore to glue them all together in no particular order, you have the Harry Potter story.
The fourth self-insert caught me by surprise when I realized it, and I haven't seen anyone else catch on yet. It is Molly.
Yup! Molly Weasley is that last of her self-inserts that I have detected. You see all of the classic signs. This character is one that has everything she wants, but money (a reflection of her own state pre-blockbuster?), and has the odd dichotomy of being at the same time well-regarded by the other cast... without in any way observable deserving it.
The character feels ownership of Harry without substantive reason, and, when it comes right down to it, I fear she actually represents what Rowling thinks of as 'the perfect mother'.
