Yep that is an option, trouble being it's nigh on impossible to get a costing in a timely manor for an urgent and significant treatment.
The insurer can tell you how much they will pay for each (code) for a procedure/cost, however trying to get all this info (codes) and what treatment/costs you may need from specialists and hospitals is problematic say for example an anesthetist who can not guarantee he will be available when required and someone else does the job but charges more. etc. Leaving you with a larger payment than anticipated.
I can see benefit from some things like a knee reconstruction instead of going on a waiting list but its not a magic bullet.
You're right, its not a magic bullet. What it gives you is the freedom of choice in those circumstances.
I wouldn't class something that gives you time to make calls and make appointments with specialists as "urgent". Getting the right codes and total cost is as simple as asking the specialist you're seeing, and checking the benefit is as easy as calling the insurer.You can get your out of pocket cost worked out the moment you leave the specialist before you even make it back to the car.
In the end, the choice you are making is not how much money you're saving from one specialist to the next but whether the cost is worth skipping the line by some degree or not.
Chances are, the specialist you're seeing isn't going to be able to fit you in for your "urgent" procedure anytime soon anyway. The exception being a dentist, which you know you're going to be paying through the nose for and realistically no "public" free option exists unless you literally cant afford to eat at which the private/public debate is moot.
The example you give around someone not being available when required is pretty slim. Medical professionals keep diaries like most, and allocate times to when they work private and public (if they do any public at all). If you're booking a time with an anesthetist as a private patient for example, if they say they are available, they're available (barring some other issue that prevents them attending, like illness, accident etc. which goes for any medical professional). The costs however between someone turning up that you booked vs someone else should be negligible in the scheme of things. Not something worth losing sleep over.