People usually get it at specific intervals as children. It's easy to plan ahead when you have everyone scheduled weeks in advance on a routine schedule. Not so much when you have everyone descending on clinics at the last minute.

The problem I think you're forgetting is that vaccines have a very limited shelf life. They can't just keep tons in stock. In a sense they're made to order, so everybody trying to get one all of a sudden is causing strain on a system that was never meant to deal with on-demand vaccination.