That may indeed be the case in the US. In the scenario I'm describing, you only have vacation days. If you are sick, then this is altogether separate from your vacation days. There is no calculation or concept of using your vacation days if you are ill, or calling in sick as some form of vacation when not actually sick...at least not structurally. In some countries, if you are sick on your vacation day you even get your vacation days back!
In what way is it better for the employee to have a fixed pool of days that illness can cut into, compared to having a fixed pool of days that is truly and only yours for vacation (while not sick!)?
Maybe from the employer's perspective, costs are lower. But fortunately these are settled matters of employment regulation in Europe, and workers' well-being has been a focus.
>have to pay for a doctors visit
Indeed it's not a perfect world...no one has to pay for this in Europe.