"Lots of misinformation" indeed.
I am sick of reading ill-informed people saying they "don't understand" why so many cyclists - having studied and assessed the risks for themselves - choose not to wear helmets, and saying things like "it should be illegal" to ride without one.
You are still far more likely to suffer a major head injury as a driver or passenger in a car than you are on a bike.
How many of you wear a protective polystyrene helmet when you're travelling in a car?
If not, why not?
Please rest assured that those of us who ride bikes (on a daily or near-daily basis) are very used to assessing the risks involved. I have used a bicycle as my main form of transport for almost 40 years, and have NEVER had a branch magically get stuck in my front wheel and cause me to fall off (or heard of this happening to anyone else - and I spent 20+ years working in bike-related jobs/ as cycling instructor, so have spent a LOT of time around regular and professional cyclists). I have never heard of anyone falling off as a result of a "sudden puncture" either. I'm sorry to hear about your dad's (extremely rare) misfortune, but don't think such anec-data is helpful here.
I know that if I were to fall off my bike my natural reaction is to put my arms out to break my fall. The chances of my head making contact with the ground is virtually zero. I'd be more likely to graze my hands and/or break my wrists (and have one friend who managed to break both wrists in this scenario). This is why I often wear gloves when riding, because they actually protect me against a real risk.
There is plenty of evidence that the heads of those who wear helmets are more likely to hit the ground, and various theories about why this is <ranging from (a) subconsciously if someone feels their head's protected they let it happen to (b) that the helmet adds extra weight, and there's gravity involved>.
When I've had a child riding on the back of my bike, strapped into a child seat, or in a trailer, I have put a helmet on them, because they're strapped in. There is a higher chance of the trailer tipping, or the bike tipping due to the extra weight of the child+seat, and because they're strapped (and could even be asleep at the time) they're far less likely to be able to react and put our their arms to break their fall.
Those of us who don't wear helmets aren't stupid. We've all given the matter a great deal of thought, and most cycling parents have given it even more thought since their children arrived, and the last thing any of us need is kneejerk judgement from people who don't have any experience of riding a bike/ towing a trailer, but as car-drivers, are directly responsible for the conditions we find ourselves cycling in.