I'm sorry things didn't go well today.
It is very common to take animals in the back for everything but the exam and vaccines (varies by vet). This is done to hopefully calm the animal down (since sometimes they don't behave as well with their owners right there). But more importantly it is done because if an animal bites its owner while it's under the vets care (exam or whatever), the vet can be sued. They just have no way of knowing who to trust and what might happen.
Hitting the quick happens all the time in vets offices even with experienced staff. I prefer to do Ellie's nails myself because I take my time and try to avoid it. They are moving quickly and stuff happens. It's really how they dealt with it that would indicate if it's an okay place. Taking him back to the back should be okay, but of course they should not give you a hard time about wanting it fixed.