My argument 'pro' would be that it is good experience to compete against other bright kids - and potentially it could be a Plan B if the state schools' 11+ was a washout and we won the lottery; and she got a scholarship; and we didn't worry about her brother being envious . From DDs point of view - it would also dilute the anticipation of the single 11+ exam which decides the state schools. For private schools - she'd be well placed to go for music/sport/art scholarships - which again seems a more fun Y6 than being all about SATS and NVR.
On the 'contra' - I would guess they would find you out at interview if you weren't totally into it? Dd might fall in love with a place where she won't go...? Exam over-load? Inferiority cOmplex? We'll sign her up regardless and finish our days in the poor-house?