Honestly, for complicated procedures such as root canal, I wouldn't risk using an NHS dentist. From what I've been told their fee structure essentially means they lose money.
Had this a few times - basically if its anything more than a filling you get the choice of the NHS botch job on the cheap or to be referred to a private and have the job done properly. Both times for me when it hasnt been a filling or extraction I've paid to have work done privately and those fixes have been good for over 10 years with no issues. You get what you pay for in the end and health isnt something you should skimp on if you have a choice (and obviously if you can afford it)
Most of the time, the private dentists and the NHS dentists are the very same people with the very same skillset, though. By and large, although my experience is mainly with the medical private sector, they don't suddenly become "better" when treating private patients or necessarily "worse" when treating NHS patients. That is my understanding, anyway.
Two weeks ago my Dentist charged me £286 for 1 hour work...I'd guess that US dentists earn a heck of a lot more than UK ones. $3300 for a days work - nice if you can get it.