[quote FictionalCharacter]“@FictionalCharacter I think that's basically the problem. In his head, ANY risk means they should be withdrawn and I think he thinks that the government is playing a bit fast and loose with the rest of us.”
I often see people saying they don’t trust “the government” about this. What about all the people involved who are not the government? The doctors, scientists, academics who have put massive work into studying this and concluded it’s a safe vaccine, the benefits vastly outweighing the risks? Are they all somehow corrupt, or wrong or mistaken? All of them the world over?!
I do feel strongly about these accusations. I work in a uni where we do Covid research including vaccine development. The researchers (many are clinicians as well) have been working unbelievably hard, so much so it’s affected their health in some cases. Then they see anti-vax stuff on SM and all sorts of nonsense about how there’s no virus, no pandemic, the Govt wants to kill people, hospitals just want to make money etc. I’ve been called a “government shill”. It’s pretty depressing.
I fear you may not be able to persuade him, but if he could learn to understand relative risk (as per the charts posted by @Wigeon) that would be something, even if he still decides not to get the 2nd shot. There is no medicine, vaccine, or treatment of any kind that is risk free. There are many, many medical interventions that are way more risky than this vaccine. No vaccine or treatment has ever had such publicity and discussion on SM, including widespread misinformation, and perception of the risks has become very distorted.[/quote]
I agree with everything here. DH has some family members who are convinced the pharma industry is keeping cancer cures away because they make so much money off the cures that don't work. I find this mindset bizarre because never mind anything else - whoever finds a cure for cancer can make zillions! But also, it's impossible for me to believe that all those doctors and researchers and specialists, not to mention the massive support network around them, are all in on some giant conspiracy to make money off the deaths of millions. In my experience in a large corporate, even just getting three people to consistently be on the same page is almost impossible! 
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So I hear you. 100%.
I do think it's the relative risk thing. And to be fair to DH, this is something he struggles with on a daily basis. He has had to work really hard because, for example, in the past, he'd be almost paralysed for days because he would have been doing something with DS and then realised that if x or y had happened, Ds could have died and he'd spend literally days agonising over it. But x or y would have been the kind of freak accident that's so rare as to not be part of a normal person's thinking. He realised he was being silly and it was negatively affecting him and he's put tools in place to help prevent this kind of catastrophising.
But I think it's a family trait. Many of his extended family, including his parents, are naturally distrustful and pessimistic. I broadly get on with his family but there have been moments where I've had to just completely ignore his parents or be actively dismissive because they will say/do/think something that's not even slightly based on any kind of real facts. Put it this way - it's a very good thing that his parents are not on social media.