Since the disease is highly age-stratified and dependent on risk factors, the same goes for the vaccines. For elderly, it's a complete no-brainer. For me, in my forties, it's overwhelmingly false and I got vaccinated as bloody fast as I could. And for anyone in a risk group, it's false as well.
But for healthy kids and teenagers? It's a wash for them personally, but if they're hanging around people in risk groups, there's a clear benefit of them getting vaccinated.
> the vaccine has been proven to be safe
There are several vaccines, and some of them have issues. The AstraZeneca one is pretty much not in use any longer in the west because of the blood clotting issue, and there are reports now that teenage boys might suffer myocarditis from the Pfizer vaccine. Incredibly rare, but the risk is not zero.
You are generally correct that the cure is not worse than the disease, for an overwhelming majority of people, but the truth is more complicated, and without long-term safety data for these vaccines, I completely understand that some people are hesitant.
At the core of the anti-vaxx bullshit is a tiny kernel of truth, and I think it's better to address that than to completely suppress everything they say, because that's just gonna make people on the fence extremely suspicious and tip them over the wrong way.
Another interesting factor at play here is that some people prefer for negative outcomes to come from inaction than action. It's like the trolley problem (https://en.wikipedia.org/wiki/Trolley_problem) when people try to evaluate morality.
In this case, they would rather not take an action (get vaccinated) if there's a chance of harm and would prefer inaction (don't get vaccinated) despite the higher statistical risk of bad outcomes.
How many kids and teenagers aren't usually around groups of 40 year olds? Are there cities which are only populated with 12 year olds? Apartment complexes exclusively for those under 18?
Kids living with their grandparents should definitely get vaccinated.