For sure, QAnon and other mischief-makers will have played in role in adding to some peoples' fears. But this is only part of it.
Many people already had beliefs that would lead them to eschew vaccines, eg those who prefer herbal medicine, acupuncture, homeopathy, etc. Who have a distrust of pharmaceuticals in general, maybe vaccines in particular, especially ones without a long track record of being demonstrably safe. This can be cultural in origin, and has been the case for a very long time, way before Covid or QAnon, or Russian bots etc - eg the preference in China for Chinese medicine, the popularity of homeopathy in Germany, etc.
For others, who are prone to health anxiety, it can feel very scary to have a medical intervention, and this again predates Covid, QAnon etc. But it was heightened with Covid due to the timescales.
It's one thing to hear about a vaccine scare (eg whooping cough vaccine in the 70s, MMR in the 90s etc), be anxious about it, and then see, over time, lots of people you know having it and being ok. This builds up an implicit trust in the vaccine. Quite another to have an illness come out of the blue and there be a vaccine less than a year later, you're being told is safe, but you haven't had chance to witness that yourself and you're wondering how that can possibly be. Eg, you're pregnant, you're told the vaccine's safe for your baby, but you're wondering how we know that for sure, when nobody's had a chance to track the development of children over their childhood and prove it doesn't affect, e.g. brain development, down the line.
Then there's the fact that we know vaccines are safe and life-saving for the vast majority of the population. But we also know that, for a very, very small minority, they cause serious illness or death. Yet we don't know which people that will be. Nobody has researched this to see if, eg. certain genes, or certain pre-existing health conditions would put you at higher risk of complications. You just have to hope you're in the 99.999% of the population who'll be fine and you'll benefit. For most of us, we're happy to take those odds. But if you're prone to anxiety and overthinking/obsessions, it can be seen as basically a form of roulette - with the odds of you getting a poor outcome looming larger than they are in reality.
For some people with anxiety, any risk of something bad happening is intolerable. Of course, this can also work the other way - anxiety disorders soared during Covid, with some going to extreme lengths to avoid it and developing agoraphobia, OCD, health anxiety etc for the first time. These anxious people were desperate for the vaccines. They also had a very skewed view of the risks they faced, but their tendency was more in line with official health policy, so nobody cares.
I work with people with anxiety, and have seen all of the above during the pandemic and since.
Has QAnon etc played into this? Yes, most likely, for some at least, though from the people I work with, I would say those sites have been way less popular than you seem to think, in terms of people going down the rabbit hole.
If it were the case that we could ban mischief makers, eg Russian bots, from their drip, drip campaigns on Twitter, here, etc, while not affecting the civil liberties of ordinary people, perhaps I'd agree with your stance. I'd certainly have preferred it if Russia hadn't been able to interfere with the Brexit vote, too! But I can't see how we can do that - not without huge amount of resource being deployed by the security services. It's outside the scope of MN for sure.
And I think we need to treat people like adults. We can counter the misinformation. But we have to allow people to speak. We also can't expect MN to act as though it's a public information services. It's a private organisation, not an arm of the state. It's a balancing act, but censorship never leads anywhere good, in the end.