The exceptions in this list are almost entirely due to possible allergic reactions. Take flu vaccines - most of them are created via eggs, so people allergic against eggs cannot use these vaccines, until especially prepared without using eggs. If no vaccine is available for your child, it should skip this especial vaccination. This also is the reason why it is so important that every child which can, should be vaccinated. If a small percentage of the population is not vaccinated, they are still safe/no danger to society due to the protection given by the vaccinated part of the population. This is not a reason to skip vaccinations for non-medical reasons.
Well, most people in the pro-vaxx community won't deny that vaccinating has risks. The risks of not vaccinating, however, are bigger in a lot of cases. I think that's the most important message in the article, that their concern might be justified, but that the coin has another side.
How do you convince your wife when your child gets the disease that vaccination would have protected them against
(which is far more likely to happen, and has cost thousands upon thousands of recorded deaths in the past)?
Your question is analogous to: "If you told you kid it's ok to go out and play, and while outside a car hit him, do you let the other child ever go out again?" probability wise.
But that's exactly what happens sometimes. Family was ridiculously overprotective of me and crossing roads because my sister got run over by a car. I was barely able to go out and do anything at all outside until I managed to get out of that environment because my older cousin got herself into drugs.
While this is merely anecdotal evidence, my story is not unique. The younger sibling paying dearly for the bias caretakers incurred thanks to correlation to a bad incident is far, far from a rare occurrence.