Too many free iCloud accounts

That is not a viable solution. One of the main purposes of iCloud is for backing up devices. Students need to use iCloud (and iCloud Drive) to back up their work in case of emergency. If they shared one user account, the files in iCloud Drive would get mixed together. Thus, each user needs to have their own iCloud account so that they can have their own iCloud storage.

Well then, how should I administer iPads in a school? If individual users are not supposed to have their own iCloud accounts and having a shared iCloud account causes issues with sharing files and quickly using up storage, is there a solution? Or is this just a situation where we have to hope the iPads are infallible and will not ever need to be restored from a backup?

If you have better suggestions about how to better administer this system, I would appreciated any constructive feedback.

Do Macs have the same restriction on the maximum number of iCloud accounts that can be created on them?

I'm worried that this would just cause the same problem on other devices (i.e. if I create a new iCloud account using another iPad or a Mac, that will use up one of their three accounts that device can create, which would lead to the same problem on that device).

The link you sent me is for Apple Configurator 1.7.1. I have been using Configurator since 2012. I am currently using Apple Configurator 2.0 as well as Apple's Device Enrollment Program (DEP) which removes the need to set up iPads through Configurator (though now Configurator 2.0 works well with DEP, while previous versions did not). Either way, both methods already connect the iPads to the Mobile Device Manager (MDM), and from there I can manage various things such as restrictions and deploying apps.

Regardless, using Configurator and/or DEP does not change anything about iCloud. As long as a device is assigned to a user, that user needs to connect to their own iCloud account in order to do backups and use iCloud Drive.