I think the problem is that it varies so much, it really can be a lottery. A lot of state schools are outstanding and don't subscribe to this non competitive business, or manage themselves in such a way that they can be competitive without it being a problem with the less sporty types.
A lot if independent schools put too much emphasis on teams/achievement which can be just as damaging, and it can be disheartening to child and parent if it's always the same children in the team, getting the prizes, etc etc.
You just have to use your judgement as best you can, hope and pray that your local primary is a good one and don't assume that an independent one is always better (although, sadly it does seem to be more often than not).
This is the problem with education as I see it, despite all the targets, reviews, etc they still can't consistently provide a good, balanced education for the children - and it shouldn't matter what your catchment is, all children should have the same chances, irrelevant of their background. Cloud cuckoo land for a while to come yet, sadly.