Is my platoon leadership allowed to micromanage appointments by requiring their approval before scheduling an appointment?

The answer is yes, Command can do this. Every unit I have ever been in had a policy letter that Command Maintenance Mondays, no appointments will be scheduled. Soldiers abuse appointment scheduling, deliberately scheduling appointments to avoid field training, maintenance, and other mission essential tasks. What Command cannot do is prevent you from going to sick call, and even THAT is tracked to show patterns.

Your appointments could very well carry over onto a day that is filled with training that cannot be shifted to the left nor the right. So yes, they can have some oversight as to when you can make an appointment. They aren't telling you that you can't make an appointment, just that you need to check the schedule first.

Depends on the appointment. Is the appointment extremely necessary and can only occur on x-date? Or, is it a standard appointment which can be scheduled so that you’re not missing mandatory training and/or deadlines? Micromanaging can occur, and at times, should be expected. Sounds like a few too many people were scheduling appointments to get out of things and the command responded in kind.