Answers
What messaging app works on a school Chromebook?
One that runs in the browser, because a managed Chromebook usually forbids installing apps and extensions. Naven, and any other web-based messenger, works for this reason — provided the school's network does not block the domain.
School Chromebooks are managed by the district, which controls the extension list, the app list and often an allowlist of sites. Anything distributed through an app store is effectively unavailable, which is why most of the messengers students use at home do not work at school.
A web app sidesteps the install restriction entirely: there is nothing to install, so there is nothing to forbid. It does not sidestep a network filter — a blocked domain is blocked whatever it serves, and that is the administrator's decision to make.
Naven runs entirely in the browser and has a help page specifically about using it on a managed Chromebook.