Your point regarding the legalities of the situation wasn’t missed. I agree with you that we need rights, but the point on the other side is while they can indeed say no, they cannot compel you to come in through means other than threatening firing you. They can say no and attempt to intimidate you, but firing and retraining for most non-entry level roles is a lot of work and often not worth the time and effort compared to letting an employee have a week off.
So while it is formally a request, we as employees can present it as an ultimatum and often if you aren’t terrible at your job, the power-tripping boss will realize it is easier for them to acquiesce and let you have your vacation than it is to go through the hiring process.