Sorry for your experience Mamaryllis 
Agree with you on the issue of supervision - that student should not ultimately have had that level of responsibility, whoever was supervising them should have been acting as a 'safety net' to ensure that things like that could not happen.
However, the reality is, that is not always the case. As a student, lack of staff has meant I have been in many situations where I should not have had the responsibility I have had. I have had mentors who have never allowed me to have any responsibility, ones who have been the complete opposite extreme and tried to get me to do things far beyond my competence eg, giving insulin injections unsupervised as a first year student (then sulked when I refused) and occasionally, a good one who has the balance right and tailors the level of responsibility/ task in accordance with my competence in that area, and supervises sufficiently without being overbearing, and allowing me to learn.
As a student, I don't usually mind at all if patients don't want me looking/ treating them. It's their choice and I'm sure they have their reasons. I find the vast majority don't object so it's not a problem. I can also understand why people wouldn't want too many students observing at once, they are in hospital, not a zoo to be stared at! And it's daft to compare a routine procedure or observation with a life saving one needing lots of hands- the two just aren't comparable.
Also a massive no to having no choice not to consent on the NHS - terrible idea, where would that sort of thinking end? Patients, no matter where they are, should ALWAYS have the choice whether to consent or not (with obviously a few exceptions in certain circumstances).
As a patient, there have been times when I have not allowed a student to carry out certain procedures. This has been due to various things for example the complexity of a procedure, how competent and confident the student appeared or what year of their training they are in. I don't think I have ever objected to a student observing, although would have no hesitation in saying no if I didn't feel comfortable with it for whatever reason. And I would never want a whole group watching, I've found that bad enough when it's been necessary (in resus and when having some of the DC's). It can be quite intimidating, and I think that should be respected.