I agree with this.
Your child has some unmet need, OP. I suspect she's finding school troubling or even overwhelming, and the fact she's eating only school pudding at lunchtime there indicates she's seeking comfort through food.
Don't panic about her refusal to feed herself. Take the long view. Don't shame her over this. Try to get to the bottom of her need to be babied. Don't make it about food because it's not about food.
I'd take turns with spoonfuls, allow her to eat with her fingers, allow her to sit at a kid table if you have one, let her sit on your knee during the afternoon or evening, tell her stories about how nervous you felt when you were in Year 1 or whatever year she is currently in, how big the older kids seemed, how noisy the lunchroom seemed, how mean the dinner ladies were, etc - try to elicit some comments from her about her experience of school /teacher / playground / her peers.
While youre snuggling on the couch, talk to her about how much she's grown since she was a baby, and how much she's learned - assure her you're proud of her and can't wait to see what new things she'll learn this year, friends she'll make, and fun she'll have.
Give hugs and high fives and extra affection for independent self help like dressing, putting on jacket/ zipping, knowing where her shoes are, helping get her bag ready the night before, and give her little jobs she can do to help you so you can praise her for listening well and doing a good job.
You can gently build her confidence so that she can spread her little wings.
Be confident that she will benefit from understanding that home is her soft place to fall.