Probably talk about it with employees like adults and try to work something out, rather than throwing their weight around?

I've worked on both types of teams, and the ones that have reasonable discussions about staffing problems are always healthier in other ways too. The ones where management just hands down edicts from on high and expects people to roll over and take it always end up being dysfunctional across the board.

If the PTO request would have been made a day in advance, very few would argue that denying it would be poor planning of the manager. A MONTH in advance though, that very much screams poor planning. Any competent manager should be able to manage and plan for that.