Vaccinated normies I've talked to seem to be using this strain of logic:
"I got COVID right after getting vaccinated, but I didn't get hospitalized or die, so that means the vaccine worked because it would have been worse otherwise."
But this is illogical, because (1) a vaccine by definition is to prevent infection and (2) there is no way to state with confidence how you would have fared without vaccination. Someone with clearer logic would say:
"I got the vaccine against COVID, but I still got COVID, so the vaccine didn't work."
And then the next step is to ask a question:
"I got COVID right after getting vaccinated. Is it possible the vaccine made me sick?"
If you're talking to someone with an illogical disposition, my experience is that there is nothing you can say to get through to them. If their disposition is still based on logic, they should be able to make the deduction without much new information.