What are the arguments for trust in the medical system, given the above considerations? — baker

Probability should be enough. Mistakes are made by people, not the "medical system". The "medical system" can have problematic bureaucracy and coordination systems in place that are prone to create problems and mistakes and these are the things to keep push for the better. The fact that many doctors and surgeons sometimes work long hours with little sleep is a sign of such structural problems.

The important thing to remember is that "the medical system" is built upon other people who struggle and fight often to the breaking point in order to help other people. The bar is low for mistakes, it shouldn't happen, but it does, because they are people. But that doesn't mean these people do these mistakes because they don't care, there are many variables involved.

and the prospect of malpractice or betrayal — baker

Malpractice is one thing, betrayal is another. "Betrayal" is a very strong word that needs some deeper explanation. Are we talking about medical staff betraying the will of the patient? There are variables there as well, the patient might not know everything about their condition, the patient might be stupid and risk their own lives. The "will" of the patient should be interpreted. If the "will" of the patient is to survive, feel better, then that is the purpose for the medical staff and doctors, regardless of what the patient believes makes them better. So a "betrayal" when treating someone could actually be a good thing that saves the patient's life. If that was against the will of the patient, why is the patient even at the hospital and not dying or suffering outside of that "system"?

If a person has already experienced serious negative side effects of a medication, or has been the victim of medical malpractice or betrayal, on the grounds of what should this person still trust the medical system? — baker

Elaborate on what the "system" is? There are thousands of "medical systems" all around the world. Some with better care than others, some with better expertise than others. So back to "probability".

Is the probability higher that you are affected by negative side effects, including malpractice, mistakes, or bad competence, compare to the probability of getting better from your illness or risk of illness/death?

Probability is all around us, the risk of anything exists always. In order to trust something, a probability has to be calculated for that trust. There will always be a risk of damage or dangerous consequences for any interaction a human is doing in their life, it's unavoidable. If people stop making probability predictions, they start operating on irrational fear and biases.

Does the "medical system" possess a higher danger through mistakes, malpractice, side effects etc. than the consequences of not getting involved at all when a health problem arises? No, it's not higher, obviously. So any malpractice, mistakes, side effects etc. that exist in the world is not enough to erase trust in the medical system. There's no rational reason to mistrust an entire system because of cases of bad luck or incompetent people.

The rational thing to do when constantly getting problems with some specific medical people is to report them because it can be that a specific doctor or medical staff is individually incompetent and a danger to others. Or an entire hospital with a bad administrative team creating systemic problems for that specific hospital.

The "medical system" is referring to the entire global medical system (as nothing else has been specified), but problems occur on individual levels or specific hospitals. These usually don't survive long, doctors lose licenses, hospitals get shut down, and so on.

The probability of getting help is still higher than any of these consequences. And people can only act on probability, the rest is an irrational belief and bias.