Do Covid-19 vaccines kill people?

For some reason strong aversion towards the Covid-19 vaccine in particular seems to be extremely common among American conservatives, and people around the world with similar conservative values.

(For the record, I don't consider myself exactly a conservative, but I do share a lot of the same values, ie. most of the core values of Classical Liberalism. So in a sense that kind of makes me some kind of exception to the rule.)

Among the myriads of (most often completely unsubstantiated) claims about the vaccine, people dying of it is among the most common. It seems like almost every American conservative knows someone who has died of the vaccine, or knows someone who knows someone, or knows someone who knows someone who knows someone, or... You get the gist.

These claims are almost invariably hard to corroborate because they are just that: Claims. Very rarely do they present any actual proof of it (such as names, obituaries, etc.)

But even if the claims are sometimes factual and true, this is still an archetypal post hoc argument (ie. "after this, therefore because of this".)

In other words: "He took the vaccine, and he died soon after. Therefore he died because of the vaccine."

The deduction might or might not be true, but it's fallacious to assert it to be true with certainty without any better evidence.

Over 30 million doses of the vaccine are administered worldwide every single day. Roughly 150 thousand people die worldwide every single day. It would thus be strange if these two things never happened close to each other to the same person, even within a day. (Even moreso because the vaccine critics often include people who died even a week after the dose.)

The vaccine is not some kind of supernatural magic shield that protects the person from death. It would in fact be quite extraordinary if not a single person died very soon after taking the vaccine, even on the same day or the next day.

But just because the person died soon after getting the vaccine does not automatically imply that he died because of the vaccine. We need significantly more and stronger evidence than just the timing of the two things.