> There is no situation where you will say a high number and they will move up to meet you if that number was greater than their max budget to begin with.

Yes, and since it's quite common to have companies where their max budget is not acceptable (probably because they are the ones who can't fill their vacancies and thus have recruiters spam everyone all the time), it's important to filter them out by saying the high number before you've wasted many hours on worthless interviews.