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

Yeah, even when you got no legs or arms and in Walter reed center. You can't be there and have existed ch 15's for being there as AWOL, even if you're in the hospital on life support or died. You still get written up derogatory as a no-show. I was at a hospital and got Ch 15 three times for a heart attack. So the answer is no you can't schedule appointments in the Army whether dead, lame, or broken dick from a terrorists attack. You're still considered Army property in 1st ID at least, and command is command.

It’s not babysitting, it’s due to others gaming the system. All you need to do is go to your supervisor and let them know the situation. Even as a Squad Leader I still give my PSG a heads up about future appointments and I would look at what was going on that week. Also if more than half the team is on appointments it just burdens everyone else to pick up the slack. E-4 mafia should take the tag team approach to details, connex life, and the motor pool.

Yes, their focus is readiness and the mission always. Managing your "appts" is not the same as denying you care. More detail is required to assess your situation and "appts." Once you become a liability rather than an asset to your unit, they would be negligent by not trying to fit you into Mission First always. Maybe you now fit that category. It's your responsibility to personally manage your situation within regulations if you plan to stay.

Short answer, they absolutely can. Should they? Probably not.
I think micromanaging has become the new normal, and not just in the military. With information being so immediately available, middle level leaders (Plt Leadership etc.,) are so afraid of their bosses seeing something even slightly irregular it forces them to manage minute details that were typically left to front line Supervisors (NCOs). That goes all the way up the higher command levels, where field grade officers are asking specific questions down the line, i.e "Why did PFC Schmuckateli schedule a dental appointment on the afternoon before a 96 and miss the safety briefing?".
The problem is, once the Battalion forces the Company leadership to do the Platoon leadership's job, they start doing the squad leaders job, and on and on.
My specific guidance to you would be to go with the flow. This too shall pass. Ask your front line leader to provide a schedule of dates/times that are blocked out for appointments.