This is EXACTLY what is needed to calm people with vaccine concerns.
It is completely understandable that people would be worried about vaccines. Unless youre already educated about vaccines, deliberately injecting a baby with a deadly virus DOES sound scary and risky!! And no amount of "just trust me bro" is going to soothe the concerns of that worried individual. Saying "the doctors know what theyre doing, just trust them" is not going to calm the worries of someone with legitimate concerns and unanswered questions.
What WILL soothe their concerns?
Information. Understanding. Explanations. Answered questions.
"This is how vaccines work in general. This is how This Particular vaccine works. This is what can go wrong with it, and this is how likely/unlikely that is to happen. And this is what can go wrong with not having it, and how likely/unlikley that is to happen." No fearmongering, no judgement; just calmly laid out cause and effect.
Have you had the vaccine or given it to your kids? Tell the worried person what your experience was.
I know someone who went to their GP during the early days of the covid vax becoming available, and she told him "I have to get the vax in order to keep my job, but im really worried about it, because its brand new and there hasn't been any long term testing on it yet. Can you please tell me why I SHOULDNT be worried." She straight up asked him that. Said she didn't WANT to be vaccine hesitant, but she had concerns, so could he please explain those concerns away.
He scoffed at her for her concerns and told her to do her own research online. She tried looking into it herself, found a conspiracy page first, fell head first down an antivax rabbit hole, lost her job for refusing to get the vax, and now is a full-scale antivaxxer. After specifically going to the doctor like "hey i dont WANT to be scared of this vax but I am so please put my mind at rest so I can get it."
Had her doc taken the time to answer her valid queries (there WERE no longer term studies on the covid vax! There still arent, because it WAS a brand new vax! It was in fact a valid concern!!) then she would not have fallen down the AV rabbit hole, would not have lot her job, and would not be neck-deep in conspiracy theories today.
Its so important to treat vaccine hesitancy with judgement free explanations and facts. Not only is it the most likely way of convincing someone to get the recommended vaccines, it can also prevent them from sliding into antivaxxer land.