Someone (or maybe several someones) on here said the topsoil from British Sugar is good - apparently it's recovered from what they wash off the sugar beets.
My soil has plenty of pebbles of all sizes in it (naturally), and it is a bit of a nuisance when I want to spike the lawn in the areas that get compacted, but the grass has been here for longer than me and it's definitely not worth the bother of taking it all up and starting again. I would say, take out what you can of the big pebbles/stones but don't obsess over it, and don't worry about the little ones.
And have a hose and a sprinkler at the ready in case there's a dry spell before the new grass gets established.
Doncaster, South Yorkshire. Soil type: sandy, well-drained
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The kids will be fine waiting to go on it. We may set up a time lapse camera to see it grow!
I live in west central Scotland - not where that photo is...