I think the "faith schools get good results" meme is a self fulfilling prophecy.
In other words, it has nothing to do with the faith/non-faith position of the school, and everything to do with the parents and the teachers? From my experience, the teachers can only go so far; the rest is down to parental support (and a child's innate intelligence)
There has been a lot of research on this.
The upshot is that faith schools on average get higher 'headline' grades, but that is due to their intake, which tends to have a higher prior academic achievement (and is broadly more affluent/middle class etc). So when you look at progress, i.e. taking into account the intake there is no difference between faith schools and non faith schools.
And these effects don't seem to be due the 'faith' element per se, but due to the fact that most faith schools control their own admissions, and the same effect is seen with other schools that also control their own admissions (e.g. academies), but not with the smaller number of VC faith schools that don't control their admissions.
Digging further it seems that once you give a school control over admission, plus you have a big pressure on schools to show great results, there is a biasing in the admissions such that the schools are more likely to admit bright kids from motivated family backgrounds than less affluent kids, less high attaining, including less with SEN. Whether this is overt or covert is a moot point, but many schools who control their own admissions (including many faith schools) have been demonstrated to have broken the overarching code for admissions which is meant to ensure fairness and transparency in the system and prevent 'back door' selection.