((Hugs)) I am so sorry you and your little one are going through this. It's heartbreaking.
My perspective is different from many others on this thread. I do recognise that some help is easier to get while a child is registered at school. I know that, morally speaking, the school system SHOULD provide for all children's needs and that it's galling to give up on that idea and let them off the hook and do it yourself when you shouldn't have to. I can see that a diagnosis is often useful, and that it's easier to get a diagnosis when a child is at school and unable to meet standard age-linked expectations there.
But what's the price for all this? Right now your child is suffering. It typically takes a very long time to get an unsatisfactory school situation sorted out. It may take years. It may never happen.
Some children's needs are such that home education is very hard work for parents. That doesn't apply to all of us. Everyone's experiences are different.
I've home educated two kids. One is now 23 and fairly average. The other is 16 and learning disabled. I actually found home ed easy, definitely far easier than putting them into the school system and dealing with the consequences, especially in the case of the one with SEN. Here are some reasons why home ed worked so well for her, and consequently for me.
Getting access to what she needed didn't depend on diagnoses or persuading anyone to give her special treatment. I simply gave her whatever she seemed to need at the moment. When she was six, she loved to play in the sandpit and dress her dollies up. She hadn't the least interest in learning to write. So I took her to the park and to children's centres. I read to her as much as she let me, which was generally limited to poems due to her short attention span. I didn't make her hold a pencil, so she didn't form the opinion that learning was unpleasant.
She needs more coaching in social situations than most kids. Because I'm there to see what is going on, I know what help she needs. I intervene sometimes: "You look unhappy. You can tell him to give your toy back." Or later I give her advice: "Remember at the park when you kept trying to start a conversation and the girl wasn't answering? That usually means a person doesn't want to talk, and it's better not to keep asking them over and over." School staff work very hard and have dozens of children to look after. There's no way they can be aware of all these things and give effective help.
Her self-esteem is high because she hasn't spent 30 hours a week with kids of her own age who can do things she can't manage. They'd have put her in mainstream school and given her easier work and extra help. On the face of it, that sounds reasonable. But what message does it give a child? It's like putting things on a high shelf out of reach of a short child and then spending all day repeatedly lifting the child up to the shelf while telling her "It's because you're too short. If you apply yourself, maybe next year you can reach it like the others."
My daughter learned to read fairly effortlessly at 14 when she was ready. She had not spent the previous ten years being told she was behind at reading and needed to work harder at it. Imagine the difference that made to her confidence. She is now discovering the joy of mental arithmetic, which she does for fun, adding two-digit numbers in a puzzle book. She can add up her rummy score and figure out how much it will cost her to buy two beds for her pets including shipping. School would have tried to make her do that at a much younger age, which would have given her experiences of failure and perhaps a lasting fear of maths. If you ask her about herself, she talks about all the things she can do, not about the things she can't do.
She does woodworking. She goes to museums. She watches documentaries about how toilet paper is manufactured. Through discussion, she learns how inflation works and what interest rates are. She does not overhear teachers telling her stressed peers to knuckle down and study because their GCSE grades will determine their success in life, while simultaneously telling HER that of course that doesn't apply to her and that it's okay that she isn't doing exams. What conclusion would she draw from that?
At 16, she still has a lot to learn. That's all right. We aren't done. There is no deadline. She isn't behind.