Unless the consultant was Superman working a second job since the demise of big print edition newspapers, he wouldn't have been able to say for sure there was nothing to worry about without an x-ray. So the x-ray was necessary.
She'll be fine.
I must have had over a hundred x-rays since infancy, covering everywhere from my feet to my neck, one with contrast and one with radioactive tracer. Plus <thinks carefully> two CTs, two DEXAs, multiple ultrasounds and I think I'm on my 8th or 9th MRI in a couple of weeks.
If I get cancer at some point in the next thirty years, it'll be down to genetics, cigarette smoke, sun exposure, pollution, the asbestos in my secondary school being removed whilst we were there, food or the long term medications I have taken - not the number of times doctors have weighed up the risks of not knowing what is going on under my skin against taking a guess that there isn't something nasty.
These days, x-rays are even less likely to be taken unless there is a definite need, as there are all the other forms of imaging, so I promise you that there will be absolutely ZERO chance of your daughter developing cancer as a result of a single, entirely necessary x-ray.