It's not that weird of a position. Penn and Teller did a show on it based on the then believed data that vaccines cause autism. (Which has now long since been discredited.)
They showed the statistics of dying/paralysis/etc were far greater than the chance of autism. So in their opinion, even if vaccines caused autism ( the debunked paper showed a tiny percent chance) , it was still safer to take the vaccine.
Even if it wasn't autism, there is a tiny chance your child can have a dangerous reaction to a vaccine. I was nervous with my children but had it done anyway. My son had a minor reaction to the Polio shot and lost the ability to walk for a couple of days.