Whether it was the vaccine or the needle, you don't force something on them like that. In this case, even if you weren't trying to point out how bad the vaccines are, they've traumatized a girl to hate doctors and will likely deny future treatment that could be critical.

This is why a lot of dentists nowadays have better numbing agents, and will give proper lengths of time for those numbing agents to set in, because trying to push your patient in and out like a conveyor belt can have dire consequences for the patient and your future customer base.