You are just nitpicking on semantics. If the total cost of publicly funded healthcare is reduced from people using helmets then that could result in not increasing what you are "making other people pay" even though you are also offering helmets at no cost. That is what most people would consider free.
People who don't use helmets for whatever reason would be more inclined to do so if they could just go pick one up, and didn't have to pay for them in a store. Even if those reasons are not that they are expensive. It's a great idea.
Well we disagree. I think there's way more effective things you can do, and this is demonstrated by the netherlands where I live. For an idea to be good it doesn't just have to in theory be net positive, many things can be net positive if you use tax money for them. The problem is we don't have an infinite government or infinite resources or time, so we should pick good measures.
All the cities maintaining a bunch of locations full of helmets for free pickup would just create more waste. I bet people would pick them up and just discard them when it wouldn't be convenient to use them. And nobody wants to pickup and wear a discarded helmet that is dirty and was in the elements so there would be huge waste. You can have a similar effect without any waste by just having a class that teaches children to ride bycicles at school and tells them the benefits of helmets and keeps helmets there for that one class. This memory would be with you for life, and you'd make your own decision.
If helmets were cost prohibitive I'd be with you, I believe in using tax money for that kind of stuff, but price is not the reason people don't use helmets.
People who don't use helmets for whatever reason would be more inclined to do so if they could just go pick one up, and didn't have to pay for them in a store. Even if those reasons are not that they are expensive. It's a great idea.