I would recommend in this situation:
Use Firefox or similar browser that has the ability to create container tabs
And open your school email in a dedicated school gmail container tab.
Then limit the container to just the dedicated sites. At least you won't leak your browsing to google any more than you have to.
This add-on is needed to do some extra features with containers on firefox https://addons.mozilla.org/en-US/firefox/addon/multi-account-containers/