Ok - there is lots of confusion here, and the usual media 'sounbites'.
Firstly, a TA or Cover Supervisor is NOT allowed to 'teach'. They can 'cover' a lesson, in the same way that I as a qualified teacher 'cover' the lesson, but they cannot plan, deliver or mark work. It IS illegal to have non-qualified staff in charge of a class for any length of time - when I was doing my PGCE some years ago I was not allowed to be in a class without a qualified member of staff. The whole point of cover sups is to provide one-off 'cover' when a member of staff is off sick/absent with little notice. As a qualified teacher, when I cover a lesson I do not 'teach' it per se, but hand out/explain/supervise the work being done. There is a subtle difference.
However, in recent months TAs and cover sups have been used, increasingly, to cover long-term sickness/absence. This is not legal. The argument that supply is no good is a weak one - most supply teachers are very adept at coming into to a school and fitting in very well, particularly if long-term cover is needed. At least this is my experience of them. They are highly skilled practitioners and this ought to be acknowledged more.
As to the issue of 'qualified' - granted, a TA/cover sup may well have a degree etc but that does not qualifiy them to teach. I have worked with some excellent support staff during my teaching career but I have to admit that they are rare. I often find them either too overbearing - to the point where they will interrupt my lessons and talk when I'm talking - or completely uselss i.e. they do nothing!
Private schools - they do employ non-PGCE qualified staff if they have a decent level degree in the specialist area in which they want to teach. However, this is becoming increasingly rare as they also like you to have some teaching experience and its almost impossible to get this without a teaching qualification.
Finally - TAs originally appeared in our classrooms with the specific role of supporting students who experienced difficulties with their learning (SEN). It is unfortunate that class sizes, the pressure on teaching staff and increasing demands of young people mean they are used more and more as extra support across the board. What we really need are smaller class sizes (esp. at primary level) and more teaching staff.