It is quite easy to do this via vaccines specifically, because a vaccine is something you have when you're healthy.
If you're ill or have a problem and you're told surgery/treatment is an option but here are the risks, then it's quite clear that you're trading off a smaller risk for a larger benefit.
But vaccines are for healthy people, and the benefit is pretty much invisible. You don't get any magic indicator showing you it has worked. So it's fairly easy to go down the line of making people wonder whether a vaccine really does work or not. It's not difficult to bring up anecdotes of Mary who had the vaccine but died anyway or Chris who got covid and was absolutely fine. Those people exist but it's larger trends we look at when deciding whether vaccines are worthwhile or not.
Secondly most people dislike the experience of having an injection and would prefer to avoid it unless they really have to. Doubly so with the tales of side effects relating to these vaccines (This surprised me actually, because I've never heard of such widespread side effects from vaccines before). Some people feel very anxious about needles. So again it's quite easy to persuade people with well... you know you don't really have to do this, or it doesn't make a difference whether you have it or not. Most people are quite happy to take an excuse to avoid something they don't want to do, the reasoning doesn't necessarily have to be that sound unless they are worried about the consequences of not doing it.
If you're healthy and you feel there is a potential risk to something then the benefits have to be pretty good in order for it to be worth it. Therefore vaccines are a very easy target for somebody who wants to open up a "gateway" doubt in someone's mind about the validity of mainstream medicine in general.