The private providers may well seem better organised but that is because they "cherry pick" services that are easier to provide. (In the same way that private postal services "cherry pick" the easier and more profitable areas of business because they are not legally obliged to provide the same service as the Royal Mail). NHS hospitals on the other hand provide a whole range of services, from relatively minor injuries to major emergencies, minor and major surgery, physiotherapy, terminal illnesses, maternity services, etc., etc. They don't just pick up the "quick fix" conditions but have to deal with anything that comes through the door.
Also, once private companies form a major part of health services, their prices will no doubt increase. We have seen this with the energy companies. There is supposed to be competition but as soon as one company puts its prices up, the others follow suit.
Recent research has established that the NHS is very cost-effective. It may not be perfect and, like any large organisation there is definitely room for improvement, but I think it is infinitely preferable to the alternative.