All my treatment (braces and bimaxillary surgery with 1 lower wisdom extraction) has been done via a university hospital dental school. In my case, the equivalent of a senior post-doc did all the orthodontic treatment planning and adjustments, overseen by a consultant. The surgery was done at the hospital by a very experienced surgeon - there were postgrads around for the surgical consultations, but the surgery itself and surgical modelling was not done by anyone 'in training'.
When I've needed to have emergency treatment (mostly ligs falling off), I've been treated by less-experienced students, but again, overseen by a consultant.
I don't have any complaints, and it hasn't cost me a penny
I don't know if they take on everybody, though. They took me on because I was an 'interesting case'.