Sorry to hear about your daughter. That's the balance that I was trying to convey. I've worked in the NHS and the private sector and have had experience of both as a patient and relative.
Sometimes, going privately can give you options of treatment which simply are not available on the NHS, or are heavily rationed.
Eg my dad has serious heart problems which make him a poor risk for major surgeries. When he was diagnosed with an aortic aneurysm the NHS advised to watch and wait. If it started leaking, or ruptured, they would operate if he lived long enough to make it to theatre. Going privately he was 1 of the first patients to have an aneurysm stented. He was in and out of a highly specialised, well equiped hospital within 2 days. Had a world renowned consultant perform the procedure and made a full recovery.
When he was diagnosed with prostate cancer NHS offered no treatment or radiotherapy with nasty side effects as again he was not fit for surgery. Privately, at a leading cancer hospital, he had cutting edge cyberknife radiotherapy. No side effects and only a handful of sessions and he is cured.
But, if I had to choose between NHS treatment at a specialist hospital or private treatment at a local private hospital I would choose NHS every time.
Sometimes going privately offers more choices with more experienced staff. Sometimes it is not appropriate.
It's so important to get advice from clinicians and research options. One big failure we noticed was not being told all options but only the ones offered locally.
There is a big difference between "all that can be done" and "all that we can do".
I wish you and your dad well OP. Don't make hasty decisions and do your research.