Sorry OP - can't really help with personal experience, but I'm in a similar situation (not for a couple of years though).
We have some friends who have sent their children to LHS and they have thrived. One is very sporty, and I think most of his social life is outside the school doing his sport - and the other is more artistic and is doing well academically. However, they are from a European country, where I think the "ethos" is largely that you send your children to the local school and they just get on and come out the other end regardless - maybe a more "hands off' approach to parenting than me (In a good way in my opinion - not neglectful or anything, obviously!). Both parents are academic and the children are all clever (the youngest is in a class with DS) and sociable - the eldest 2 are at, or planning uni.
Other friends are also sending their eldest DS there next year - they did have a look round at a few options this time last year, and thought the ethos of the school was pretty good, pastorally. Their DS has mild dyslexia and they thought the school was quite well set up to help with that.
They also considered another school - Beeslack at Penicuik. It's in Midlothian so out of catchment but there is a direct bus from Liberton and there seem to have been no issue with placing requests. There are two other families with high school children in our street, and they have both sent children there in the past 5 years. I think they felt it was a better fit for their children than Liberton - but I suppose it depends. The friends above who chose Liberton over Beeslack thought they felt similar to each other, but their DS wanted to stay with his friendship group and this was more important to them.
I have to say, personally I'm very uncertain - but I haven't really looked into the options in any great detail. I know there is a new school planned - but that won't be ready for 4-5 years, so after DS has to move. The academic results aren't great, obviously - and to be honest I'm not that concerned about that. I think a motivated child supported at home will do reasonably well academically anywhere - definitely not ending up a "failure"
. But I do worry about the attendance/ destination stats and wonder if the "aspiration" side of the ethos is quite right. I think a lot will depend on the nature of your DD, and if she is bookish and quiet, and has a bunch of like-minded peers, I imagine she will be fine.
I have had thoughts of going private - though to be honest, we can't really afford it, and DH is refusing to consider it other than a knee jerk "snob" response. It's not a route I'd ever thought I would be considering, and I do have reservations in principle. And I'm not even sure it would be better for DS! DSs primary school do lose some children around P5 to the private sector, though not loads, and luckily none of his friends have gone. He does have quite a "nice" friendship group (yes, yes I am a snob) with parents who are having similar reservations and putting out the same feelers as to where people are thinking of going for high school! So I'm hoping things will be OK.
Anyway - in short, I'd like the same reassuring stories as you...