DS goes to one of these schools and I visited pretty much all of them when we were viewing schools. We needed tribunal to get his placement, but I know from talking to other parents that not everyone has been through tribunal: sometimes the LA pulls out ahead of the hearing, or in a few cases the LA has approached the school, as you say. And yes, it's often because the child's needs are so complex that they've been turned down by every other school in the locality.
All the students have a diagnosis of AS or HFA, but most have additional dx as well: often ADHD, SpLD, dyspraxia, some have epilepsy - very common co-morbids with ASDs so not surprising at all. AS is the main obstacle to their learning though, and it has a knock-on effect on all aspects of their education and independence/social skills.
Academically, it very much depends on the student. Almost all the students have above average IQ, but not all of them can demonstrate this through academic work. So some students fly through with A grades, but others take vocational qualifications. The small class sizes means that teaching can be modified according to the needs of the student very well. There is a general ethos that life skills are more essential than academic qualifications though - so the number of GCSE subjects is less than you'd take in a mainstream school. The HT said they often have to pick up students who have gone through grammar-type systems where they've survived due to the highly academic structure, so they come out with strings of qualifications but absolutely no life/social skills and then fall apart with poor mental health, so I'd be very careful about looking at schools like that.
DS had challenging behaviour when he started at the school but this has dropped to almost none at all now - most students with AS only display that kind of behaviour when they're in the wrong environment. I'd say that behaviour is excellent in the school, but it obviously takes time to manage it. But not all students start off with challenging behaviour, sometimes it manifests in extreme anxiety, or self-harm, or inappropriate behaviours. But all of this comes from having unmet needs due to their AS, it's not something that can be separated from that dx.
£130k sounds like a shedload but it really does go on the staff - keyworkers dedicated to each student, TAs, lots of direct therapy, highly qualified staff, ASD training for all the staff (even the gardener/cleaners), access to psych services, and activities off-site (which has a social educational value, not just for fun). They don't have luxury facilities or unnecessary extras.