The accurate answer is 'it depends'.
If the SALT is in part 6 of the statement, then the LA are not responsible. They are merely stating provision that is given by another agency which supports the child.
If the SALT is in part 3 of the statement, then the LA are responsible for the provision. They will say that they contract with the NHS, therefore the NHS provides it, but in truth, they have to ensure it is provided, and if the NHS cannot do so, they must do so privately. But you are likely to have to go to tribunal to make them.
My only experience with MS with DD was MS preschool. It didn't work, even with 1:1, because the expertise was in the professionals who visited to advise, and the assumption was that the preschool staff would somehow understand the techniques that were suggested.
The reality was that the staff didn't understand the basic premises which made the techniques work, so couldn't adapt the techniques to other situations, only the specific one they'd been taught and had 'modelled'. Because they didn't understand the basic concepts sufficiently, the staff assumed that the visiting professionals had simply misunderstood DD1's difficulties
and therefore disregarded their advice.
For example, using 1 key word. The staff member said 'DD1 can understand much more than that', so disregarded it. No, DD1 is brilliant and smoothing over a lack of understanding with social manipulation, so appears to understand, but the effort of that causes frustration, hence the behaviour which the staff saw as the priority. Had they listened, they would have negated much of the behaviour by reducing DD1's frustrations.
As you have found, it seems that in MS a lot of the strategies involve dealing with the child who is causing an issue, which gives that child massive attention, and reinforces the behaviour. For example, if DD1 didn't want to do circle time, the staff were told to take her off and do something she wanted. So, she got to know that all she had to do is throw a tantrum, and she could play with a balloon. At Special School, they said 'Oh dear, DD1 will find the first few weeks a bit tough then, because if we do circle time, she'll have to actually do it...'
It takes a lot of skill to successfully support a child with SN in MS, and the vast majority of SN assistants are hired with very little experience. Then, they get experience of one child, but think that they have 'done SN'. That's why I love SS. The staff there 'do' 110 children with SN daily. In the classroom, in the playground - it's all teaching time at SS. The children aren't just left to it at playtime, activities are set up with the intention of teaching a skill. The children don't have to use them, but they are there, and the teaching staff are all out on the playground (they have two lunchtimes, so that the staff are staggered).