Edited to add: The issue in this story is that someone is spreading false information and people are acting on it and people are dying. Truly sad.
This is a bad argument and does not work in practice, because people perceive it as a language trick. If a child was vaccinated and then died in a result of the process, most people call it "die from a vaccine" independently from the mechanism. Trying to make the risk of the process look smaller than it is by focusing attention on technical details makes conspirologically minded people who do not understand the details even more suspicious.
In other cases you mention people go to doctor, because they can clearly see that not going is way more risky. So a better argument is to remind them that if enough people choose to skip the vaccine the danger from the virus will be much higher than the danger of nurse making a mistake.
Of course things can go wrong in medicine, but that's not related to vaccines. If this is the kind of concern you have, you should never allow any doctor to put a needle in you, because it might be contaminated with something. You also shouldn't take any medications prescribed by a doctor because the pharmacist might mix them up with something. You certainly shouldn't ever have surgery, because the surgeon might misread the chart and perform the wrong procedure. Of course errors are possible in medicine, but that doesn't mean that needles/medications/surgery are inherently dangerous, it just means that human error happens. If you choose to never use healthcare for that reason, then you're wildly irrational (which is fine, just don't harm children because of your wild irrationality).