It’s currently limited to the way you describe above. Most orgs I know limit the ability to create Office 365 groups which restricts the creation process and therefore the creation of Teams. The global admin would then create the Teams with them as the owners.
It would need someone to look into but a flow may be able to automatically add the global admin as an owner if you create them via Forms, for example.
I would submit a uservoice if you want to see it introduced. The reason I think that Microsoft didn’t add this is because it would be like admins having access to all mailboxes. It would mean that nothing is private to users in the organisation. However, I can see why this is required particularly in environments like education. A high level of moderation and safeguarding is needed.
Thanks. I am currently trying to educate users to put files in the Class Materials folder. But I want to check they are doing it properly, and for that I need to be an owner of all the Teams. Nightmare admin task and not something I want to do.
Understood! Are you the actual teacher of all those classes - if not, may be worth training out the teachers, support assistants or the children themselves (self police) to be able to check this to reduce the admin burden.
If you are only needing to see files you can make yourself site collection admin to the attached SharePoint sites by using the set-SPOsite command with -identity “siteurl” -issitecollectionadmin $true flag on. This will give access to files only without having to show up as being in the Teams.