Actually, many, if not most can and do. There are times of the year when it's all hands on deck and PTO is subject to supervisor approval. If staffing is insufficient, a supervisor has the discretion to reject the request and ask to resubmit for different t dates. Happens occasionally where I work.
guess i should reword that. they "do" have the say so to approve. but in my 17 years into my career. i've never once been turned down and have never heard of anyone at any of the 4 places i've worked being turned down.
i'm also hourly, i'm not stupid enough to take off when we're busy and miss out on OT pay just to take off lol