I agree, paying for regular small things doesn't fit into the insurance model.
However, I think that it would be handy in terms of prevention. It would be like investing in the health of the person. Perhaps if I called it a subsidy rather than insurance, it would make more sense. It would raise the price of insurance by the amount you suggest, but only if the vaccine had no effects. If encouraging everybody to take the flu vaccine lowered the incidence of serious complications, then the total cost for health insurance might go down.
A similar example would be a policy that my car insurance company had. If new drivers took a class or kept a log while driving then their rates would go down. In that case they were lowering costs by trying to make people better drivers. In the case of health insurance, encouraging people to live more healthy or get vaccinated (for example) would lower the chance of serious complications.
Overhead is a problem though, and I don't know how much of a hassle it is. I'm young and thus haven't had to deal with health insurance at all. I would think that if a provider guaranteed that everyone they covered could get a vaccine, and had an efficient method for the vaccine distributer to check that it was covered (e.g. a credit card sort of deal) then I think that overhead could be minimal. I have no experience in this regard though, so overhead could mandate that small expenses shouldn't be paid for.
Summary: I was taking more about health care by subsidy rather than insurance.