Nope.
There are lots of variables; some alluded to on here.
'Your choice of doctor'. What criteria are available to you to choose? He seems 'nice'? Some of the best surgeons I know have the bedside manner of an angry ox.
And always seeing the consultant? (Who is often 9-5 NHS, anyway)- the senior reg is often rather more up to date in current practice as he needs to be.
Quick 'cancer care'? No, you'll get a 2WW on the NHS. Go with that.
Legally, you can't dip in and out if the NHS. Yes, it happens, but it shouldn't. Go private for something? Stay private for its duration, 'complications' and all.
Enough nurses? Probably. But, and I am not speaking for all private hospitals, but where I am, Hants, the local private hospitals are staffed by ex-NHS staff towards the end of careers who don't have the stamina or ambition for NHS work.
My point is- it is so rarely 'worth it'.
I'd PAYG, personally, as needed. Not plough hundreds into schemes that rarely cover what you might need.
Car crash? NHS.
Cancer? NHS
Paeds? NHS
In other countries (Germany, Australia), private is far more closely aligned to state; 'ordinary people' pay private. This helps keep it honest.
The 'honesty' in British private health care isn't there.
I have been asked to privately CT scan the brain of a young urban professional man who knew aliens had embedded electrodes in his head. To prove to him none were there. I rejected £50 in refusing.
Someone else shrugged and did so. Funnily enough, no electrodes.
Patient sneered and said 'Yeah, but they're invisible, obviously'.
Why was he not (privately) referred to psychiatry? Because there was a buck to be made.