We have already implemented a prototype for such a system, which works by Shibboleth login: Frugal Sheets. An engineer at Télécom ParisTech is currently looking into this, because this prototype would have to be cleaned up, extended, and maintained to fulfill its role. Particular emphasis would have to be put on the security and traceability of the data, and on the ability to import and export it.
A detailed case for the Frugal Sheets is made here.
A detailed argument for a shared calendar is here (“Bologna Talk”). A shared calendar proposal has been in place every year for Paris-Saclay since 2016.