I would agree with a few of the posters above. Have taught just under a decade in state primaries and more recently a few years in indi-prep.
From my experience, in prep's the parents pay for the screening and ed-psych. The prep has a few EP's that they know of and have worked with, and the parent makes the decision which to use. Teachers and parents are both able to flag up concerns and keep an eye on things, before recommending a screening if that nagging feeling that something is up doesn't go away after a few terms. IEP/IAP's are very well organised and reviewed on a regular schedule between home and school. I have seen real improvements in dyslexic pupils. The parents also have to pay for the 1:1 individual support if they opt for it, all of them do. This is usually a weekly hour long 1:1 lesson with a TA. I haven't yet seen it materialise as 1:1 full time in class support for a specific pupil. At that point, I imagine the cost is too high and the state would serve them better (as they would then have a statement to say why that level of support was necessary).
In state, the SENCO is overwhelmed. Often children present as middle-ability and muddle through, but if you looked harder you might realise they were supposed to be higher ability. Only the lower ability pupils are pushed for screening as the school has to pay for it (it's expensive!). I have seen middle to higher ability pupil's parents pay for it privately because the school refused to screen, as they were generally 'fine'. This means that twice exceptional pupils are missing out, (exceptional because they are both of a higher ability, AND struggling with a specific learning disability).
I would agree that the state sector deals with more complex needs much better, but for dyslexia (and similar dys-) the children get a much more individually catered and intense level of support. In the state, so many have these 'simpler/softer' needs that they realistically don't get the time/support, as those with more demanding needs take priority/more time.
It's such a shame that those who can't afford private yet 'only' have a 'small' learning need don't get the support they need in state, but with so many in one class it's a huge job for just the teacher to manage day to day (again, the SENCO won't be there day to day to practically help in my experience and the TA is often with the higher needs pupils). I don't know what the answer is.