House we found dd's tutor by word of mouth - but we found her because dd was having panics about maths in Y4, so it was purely to boost her confidence. Nothing to do with getting in to anywhere, at the time. She had a term of tutoring (which she enjoyed) and her confidence soared. Then we went back this year to talk to her about secondary schools. Do pm me if you want contact details (though please, not anyone else, because last time I offered I had floods of pms!) as I can't be doing with the secretive stuff - its total nonsense 
Dad, I know what you mean and I feel the same about stress and pushing - I really do. Odd as it sounds, so does dd's tutor, though I guess its her job to get kids to where their parents want them to be, a lot of the time. Sadly. On the plus side, she says she won't go along with the parents who want under 7s tutored intensively.
In our case, we felt taking a slow leisurely route to get the curriculum covered and confidence up was the right way for dd - she doesn't do 'intensive', its more supportive than anything else. But I think we're unusual in this, according to the tutor.
dd has about half an hour to an hour per week of school homework, and about half an hour of tutor homework. She does three after school activities, one of them only 45 minutes. And she has made a dent in the sofa where she likes to sit. Trust me, she is neither stressed nor pushed 