We have it through my husbands job and it has been a godsend, I totally recommend it if it would suit your circumstances, but like Marlespo said, only if it pays for what you need.
For example: I just had an arthroscopy done at a beautiful private hospital (own en-suite room) nurses changed shift every 4 hrs and the total cost to me was £148 and I was in and done within a week for something I would be on the NHS waiting list for over 10 mths.
Husband had a similar ankle surgery, his excess was £200 and he was done within a month instead of an 18 mth waiting list- same surgeon of course!
The only thing I would say is that the billing process gets quite confusing as they pay portions of each bill (anaesthesia/surgeon/etc) and then present you with the excess, but then as an American, I presume you will be kind of familiar with that process.
Aftercare, second to none. My surgeon got back to me with a query at 8 pm at home, that would never happen on the NHS, you would need to go back via your GP for a referral to the appropriate surgical team at the hospital.
Small note on cleanliness: I have never seen such a clean hospital, each nurse/surgeon etc scrubbed hands with the cliniscrub and then alcohol gel before even doing BPressure every time they entered and departed my room (room which had room service/newspaper/electric bed/balcony/Sky tv- drugs and vitals were monitored every 2 hrs and I would have happily stayed there a few weeks
With the way the NHS is today, I would definitely give it some serious consideration- but pre-x conditions won't be covered in many cases.
Hope that helps JD