Files have never been backward compatible to previous versions. ZBrush 2021, for example, was not able to open files created by ZBrush 2022.
You said you were hoping you could stick with 2022 for the next while. You can. Any student that gets an academic subscription is not forced into ZBrush 2023. They can instead choose to install ZBrush 2022. You can thus all be on the same version.
Regarding deployment, while the Maxon App must be running on all lab machines, it does not have to be used for deployment of ZBrush. The ZBrush installer can be downloaded from My Maxon and then deployed using the same silent installation systems that you have used for previous versions.
I hope that helps.