A mindset where we're incredibly worried about miniscule risks but mostly oblivious to larger everyday ones is ignorant and dangerous.
Exactly this. People's ability to assess risk and make sensible decisions based on facts, not feelings, is very, very poor. Every day we make thousands of decisions without even noticing that's what we're doing. Crossing the road, getting on a bus, driving the car, buying a prawn sandwich, chopping an onion - we don't even notice we're making decisions here, but we are, and they all carry risks. We ignore them because ordinary life becomes impossible if we think about every single detail. Habits become established and we just do things without thinking about them, and ironically it's when we stop thinking about the risk of everyday things that we can drift into very unsafe behaviour - not checking both ways before crossing the road or texting while driving, for example.
When it comes to out of the way decisions that we don't make routinely, and where we're aware that there could be catastrophic consequences, some people massively overestimate how likely those consequences are to happen to them. It's all too easy to visualise the result if you do have the one in a million bad luck to be affected, and all too easy too to ignore the harm that could follow from your evasive action, which is far more likely to happen.
Example 1: you don't have your baby immunised against measles, although there's no medical reason not to. Your baby avoids the very small risk of harm coming directly from the immunisation, which is what you're frightened of. Your baby is, however, now at far greater risk of getting measles, being miserably ill for a week at least and having quite a high chance of nasty side effects and even death. Your baby could also infect someone who can't be immunised for medical reasons and cause them great harm too.
Example 2: you don't send your child on a school trip to London because you're worried about terrorism. Your child is indeed kept safe from the vanishingly unlikely chance of another attack. However, your child misses out on a fun experience, on the learning objectives of the trip and (most importantly) learns that the most important adult in her life does not make rational decisions based on evidence and lets anxiety rule her life.
I grew up with a very overanxious mother and it's affected me. I'm very risk averse and I do worry unnecessarily about things. I try to overcome it, though. I would like to be a strong, resilient person and to make proportionate decisions based on logic, not fear, and that's what I want for my children too.