without knowing if you have 10 or 20 days off per year, why not just ask for it?
I think there are three main reasons.
Two are cultural differences. First, a good part of US culture is still grounded in the puritanical attitudes of the
colonizers founders, and a big part of that was work before play. Second, the US' political economy is farther towards the private end of the collective/private spectrum of capital, and so one's "value" as a person here is more closely tied ot their economic productivity than it typically is in the EU.
The third is more pragmatic. We have a much weaker social safety net and things in the EU that would be provided by the state are provided by one's employer---this includes (some of) one's retirement income, and (more importantly) access to affordable health care. If one loses one's job, healthcare costs become an immediate and major concern. So there is a (largely unspoken/unacknowledged) fear: "If I take too much time off, I'm going to be in the next round of downsizing, and I can't afford to lose my health insurance." Even the state-provided retirement income is contingent upon one's employment history and earnings record.
My daughter's health insurance was through the University of Wisconsin; while she was a graduate student instructor, her monthly cost was quite reasonasble, as the University subisidized it. She graduated in December, and has been on the job market since then. During that time, she has had to pay the full cost of her insurance policy, and will until she starts at her Assistant Professor gig this July. Luckily, she has access to financial resources---her mom and I are picking up the tab for the part that UW used to pay for. But, if she did not have access to those resources, she would have spent that six months uninsured, paying the full cost of any necessary health care out of pocket. That's a gamble that thankfully, she did not have to take.
I'd have a lot more to say about all of this, but this is not the controversial topics board.