I disagree edam. For example, "the weirdo?s who were allowed to run schools under Tony Blair" were not normal businesses subjected to the exposure of the free market. Those schools were set up by wealthy individuals (in many cases with more money than sense) and as such they were run as the fiefdoms of whichever nob-head happened to be in charge. If your kid went to one of these schools tough. If you don't like the idea of evolution being banned in your kid?s school what could you do about it? Nothing. You certainly couldn't take your money to a rival supplier. The only people who thought that not government owned = free market were the governments of Major and New Labour, hence the colossal balls ups with everything they tried to privatise (see British Rail, Royal Mail, Utilities etc...).
In a real free market system if you don't like the quality, price, ethics, location, colour of the wallpaper etc., you can take your money and go elsewhere and be provided with an identical service with a rival supplier. This is why Tesco's is in competition with Sainsbury?s whereas South West Trains isn't in competition with Merseyrail.
"It's all very well to say it doesn't matter who provides public services, but the evidence is that it does. For both financial and other reasons. The taxpayer often gets a very bad deal" I can see where you are coming from here but more often than not the big financial overspends are caused by the government using a piss poor procurement system and then changing their minds every five minutes. Consider the new aircraft carriers for example. Is it BAe's fault that they are going to be expensive for the taxpayer, obsolete, unnecessary, and will be immediately mothballed once they are completed, or does the blame lie with whoever allowed the 'no cancellation' clause to be written into the contract and has since changed the finished designs several times. Likewise, PFI is a useful method to get desperately needed infrastructure built, however, it was never conceived to be rolled out on such a grand scale.
You also state that the private sector has no experience providing A&E cover. I beg to differ. How do they manage on the continent? Just because its foreign owned doesn't mean that they don't know what they are doing.
As for your final point about motivation, this is again, wrong. I couldn't give a frogs fat arse what makes a nurse/doctor/bin man get out of bed in the morning - so long as they are motivated and do a good job. To turn this point on its head, why should we be allowed to get away without paying public servants what they are worth and, instead, make up for their piss poor pay by abusing the warm feelings they have (however misguided) towards their fellow man? If a Tory came on here and said "let's cut nurses salaries because they're all nice people and are more motivated by their desire to help those suffering than they are motivated by money" they would get torn up for arse paper - yet, how does this differ from your point that "people providing those services should be motivated by the ability to care and/or to do a good job, not by mega-bonuses".