I've been in a similar situation but as someone authorising leave then having to fight for it higher up. It's not as bad as what you describe but exceptions happen and when they do it is very stressful to have to fight for flexibility against the fear of setting a precedent. I wouldn't say it's entirely unreasonable but it does cause additional stress, frustration and rub the employees the wrong way. Many companies are like this at some level.

The procedure I follow is that if someone gives the right notice it is almost certain it will be accepted. I've never had to turn it down and I'm not even sure about the specifics if it were to be turned down. The idea for the notice is to be able to make adjustments for the lack of availability. If it were turned down there would have to be a really good reason for it.

When people do give short notice there is a much higher chance of it being rejected particularly because the time will not necessarily be enough to change plans, schedules, reallocate resources, etc. There isn't an absolute way to go about dealing with this. Personally I only become upset if notice was not given at the earliest convenience. If the absence does not cause disruption then I don't see the point to create disruption for the employees under me for not particular reason. My approach to management is that it does not make sense to antagonise those who work for you.

This can improve mutual respect and attitudes. The other school of thought however is that if there are too many exceptions and the policy is too permissive people will get into a habit of taking and being able to take short notice leave.

I am not sure where the best balance is. It's not always easy to make a perfect policy for handling exceptions or to discipline people for making too many exceptions.