I think you're right to take him out of there.
But I really, really urge caution about calling it HE. Obviously you can ignore the following advice completely - sometimes you just need to go with your gut. But it's advice from someone who has been stuck in the mire of local authority shenanigans for the past 4 years.
If I were in your shoes, I would say that this school is clearly failing to meet need - they acknowledge this themselves. However you have evidence that your DS is perfectly capable of learning and behaving (take some videos of you and him at home). You do not feel he needs a special school at this stage and want to exercise your parental right to express a mainstream preference.
So you would like to line up an alternative mainstream school BUT he will need transition into that setting to be very carefully managed and gradual.
In the meantime you will begin tutoring him at home but you will need to agree a personal budget in order to find some other tutors and perhaps someone in an advisory role at a later date (e.g. to advise on school integration).
Then just start HE as you were planning. You can take your time about visiting schools, researching home tutors (it is good that he can generalise his good behaviour to other people - this is best started at home before transition into a school imo). It could take you years while you are happily home schooling.
But at least this way you haven't let the local authority totally wash their hands of any responsibility. And the effect is exactly the same but called 'education otherwise than in school' rather than 'home schooling'.
At any point in the future, if you have just had enough of the whole thing, you can then say, forget it I'm home schooling. So you have lost nothing. But you might gain valuable resources and support for the future if you do it this way.