If you're looking at StH, have you considered OLA?
Family members go there (having not got into more academic schools) and really like it. They're not Catholic.
I do think you need to consider that if your DD is not academic she won't be getting the wonderful A level results which the top end of the very academic schools produce even if she does get in. Far and away the biggest factor is the individual, not the school. At that point it's a decision as to whether she'll do better as the least academic person in an academic school or as the most academic person in a non academic school. (And of course there's a big spectrum in between).
If she's average in intelligence on a national scale, she isn't going to get into StH and the like anyway - their bottom end is way above national average. Average nationally is about a GCSE grade C and these schools have almost nobody who gets even one grade that low.
I didn't like Headington at all to the extent that we didn't apply. Their attitude to "what about anorexia etc." was very much "oh, that isn't a problem here." OHS's was "we know this can be an issue, so we do this and this and this and this..." Though DD is now at uni, so this is a few years ago now.
OHS has this reputation as a hothouse. We didn't find this at all. They were incredibly supportive of DD's (non-trendy, non-school) sporting activities and the fact she was prioritising them over absolutely top academic achievement. And you've got to love a basic plain navy uniform with minimal faffy embroidered bits. Even if you buy it in the approved shop (we didn't after year 7) it's half the price of the Headington one tops.