I am a dentist. I have been to Birmingham Dental Hospital. It is hardly a social venue, and I can appreciate that lots of people who attend are anxious of both dentists and hospitals. It will never be as nice or efficient as a private clinic either.
But it provides a free service. Having a general anaesthetic (being put to sleep) privately would cost in excess of £1000 for the theatre and the anaesthetic and recovery staff; add to that the actual cost of the surgery. Having it done under sedation (sometimes confused with being asleep) in a clinic would still be several hundred pounds privately. Try to keep that in mind as you read the rest of this.
The BDH is a teaching hospital, and students are around. Dentists have to be trained somehow and observing a senior colleague is how you begin. When a dentist would like students to observe they will always introduce the students and ask the patient if it's ok. It's hard to see this not happening. Even if they don't, it's not beyond the wit of any patient to ask for the students not to be there and the staff have to respect that. You should have made your wishes known.
Second, the way that Day Surgery works is the same up and down the country. All the patients for the whole morning or afternoon arrive at 8am or 1pm respectively and are made ready. The surgeon will come round and greet them all. Once the list has begun, the surgeon doesn't leave the theatre or stop working. This means that one patient will have to go last and wait from 8am until gone 11am to be operated. The reason is that this is the most efficient way to use an expensive operating theatre and surgeon. If you have patients coming in 20-30 minutes before their actual operation time then the surgeon has to come out of theatre for each one which will probably take 10 minutes each. Do that for 4 patients and that's 40min lost, which would otherwise be enough time to operate on a 5th patient. If there is, say, a missing x-ray then this will cause a huge delay to the whole list in this situation because there is so little time before each operation.
So if the operating list worked like you want it to it would be at least 20% less efficient. There would need to be extra lists, waiting times would rise, and the cost to the taxpayer would shoot up. This is why you come in early and have to wait. Of course, we all recognise that this is a big downside for patients but this should have been explained to you. If it wasn't, well that's a disappointing failure in communication. But surely not worth 1 star!
Thirdly, I think you may not be comparing like with like here. I'm guessing you had 4 wisdom teeth out while asleep, on the NHS. I doubt you'd have been referred so soon after for another 4 teeth out. I'm guessing you had one out, or maybe some repair work to your teeth done privately. Hardly a fair comparison!
I think that your review is deeply unfair to hard working NHS staff who almost always provide a very good, and free, service.