That's just because the manager is then responsible for the work when people take vacation, so dissuading it as much as possible saves them time and stress personally.

Vacation is a big problem of you're running with just enough people to operate the business, which most are people are one of the most expensive parts of the business. Unless you're not doing anything time sensitive, mission critical, or customer facing, then being on vacation means someone is stretched to cover you.

Thinking that everything is time sensitive is often an illusion or management failure in my view. You can't be constantly on the critical path.

It's not real. I've gotten that when asking for more money too. Running a guilt trip on employees or potential employees is a pretty standard tool of psychological manipulation and intimidation.

A manager whose bonus is on the line and who knows that they have no chance of getting an increase in holiday allowance approved is quite likely to do that.