This seems to be more of a problem with insurance - there often is little transparency and doctors simply charge all sorts of things to the insurer that the patient has no idea about.

I think that is one of the main challenges of health insurance.

In my country (Germany) I have actually shied away from switching to private insurance, because they pay doctors more, and doctors are known to do as many tests and procedures as possible (not always a good thing, but in the end it is a gamble - too many examinations/procedures or too few?).

The "public" health insurance is quite regulated as to what gets paid and how much. The downside is that doctors have to waste a lot of time on bureaucracy (every examination has to be entered into the computer to determine the costs), and transparency is still missing (patients typically don't even know how much their treatments cost). And some committee decides what procedures get paid. I don't trust committees. I trust the market (I mean I trust "in" markets, not necessarily any given market as markets can be broken by regulations, lack of transparency, all sorts of things). I have also spent a lot of money on things that weren't covered by public insurance.

One story I heard is that for example in the UK they won't pay for dialysis after a certain age. So public health insurance is not an automatic solution to every problem.

I think it is actually unavoidable that insurance will have to make a cut at some point, because medical procedures can become arbitrarily expensive. Perhaps even now for the right money, people could be kept alive indefinitely.

I would have no problem with a doctor trying to be profitable - just as with other businesses. I guess the classic assumption is that to be profitable, businesses have to provide good service.