MY dd got in to a pretty prestigious school at 7+ from a not-amazing state primary and 3 yrs on is thriving there. I think I've been on threads on similar subjects before. She failed to get into another school, and in hindsight it would have been totally wrong for her, so I very much believe the right school finds the right child.
She wasn't tutored. Nor is she remotely a genius. Very shy, no dazzling extra-curricular activities to boast about. In preparation, I did a couple of VR tests with her (she was naturally good at these and I suspect that's what got her in) and made a pathetic attempt to have her learn her 2- 5 x tables, not sure she did in time, she still doesn't know them that well.
So I firmly believe that tutoring isn't necessary, in fact I believe the industry is a huge con. The schools do take the educational backgrounds into account and they can see right through a child who's been intensively tutored, In any case the heavily tutored children I know often failed, despite being considered dead certs, and I think it's because all the pressure was too much for them.
The two years ahead thing is bollocks. Ahead of what? Probably ahead of the lowest achieving child from a very difficult family background in the state school. There may be one or two gaps, but again the schools take this into account.
I don't know about CLSG, some of these schools don't ask for a reference from state schools but others do. How the current school will react to this, you'll have to gauge. Mine were extremely gracious about it, though clearly they'd rather dd had stayed.
I hope this makes you feel a bit better. Oh, and I'm very glad I sent dd to her school from a state school, as my friends at the current school have told me blood-curdling tales of the madness surrounding 7+ at the pre-preps. DD would have hated that and i'm sure it actually worked to her advantage to avoid the circus.