I'm happy that your daughter received a good level of care, but you should be able to expect that.
The NHS is, I'm afraid, absolutely terrible. The UK have done a fantastic job of gaslighting the British public into believing they should be grateful for it, despite the extraordinarily poor level of care that is given
I have rheumatoid arthritis. On many many occasions throughout my teen years I had classic symptoms including excruciating pain and loss of mobility. I was never given so much as a blood test, I was accused of lying to cover up for "not having revised for my Alevels", I was told that I was to "stop wasting appointments that people who are actually ill need" when I cam back in my early 20s as a single mother and was told I was "just tired" it was incredible, and it wasn't just one doctor, it was multiple places in Manchester. Birmingham and Cambridge (born in one, studied in the other 2).
In Spain, I had one bad flare, went to the GP, blood test the next day, specialist rheumatology appointment within a week, within a month I was prescribed a biologic that changed my life. If I still lived in the UK, and the treatment had continued as is was (and I see no reason why it wouldn't have) I would have permanent damage and probably be in a wheelchair by 50.
My story is not unique. The maternity care in the UK is dangerously poor, the fact that people can barely get a GP appointment is absurd (here I live in a working class area of a major Spanish city and if a child has a fever and you call the GP at any time between 0800-2000 Monday to Friday you will be given an appointment with your paediatrician the same day and usually within an hour or so. If outside those hours then there are out of hours services. Same for an adult
I could go on and on and I'm sure there are good doctors and nurses in the NHS but as an organisation is is dangerous underfunded and routinely fails to pick up on complex illnesses because of the dismissive attitude that is often given towards patients and I suspect this is partly due to lack of funding, burnout etc but also due to a culture where noone dares criticise -you only need to look through Mumsnet for a day to see multiple posts of women who are clearly unwell wringing their hand about whether they "really need" medical attention, because there is this culture of "you need to be half dead to take up one of our valuable appointments " it's extremely sad.
So, I'm sorry your daughter broke her arm, the poor thing must have been in a lot of pain and I'm glad she was seen quickly. But this is not an illness requiring a diagnosis, it's a fairly simple and clear injury that was fixed. As it bloody well should be! A child with a broken arm should always be seen quickly and taken care of - that's basic.
But the NHS, is still, unbelievably shit.