Thanks for the reply, suteja!

This is utterly bizarre, but it started working fine the next day, with no changes at all.

But to address your points, just for anyone else in a similar position to refer to:

• The credentials were fine. Files and folders were being pulled from the repo. But clearly something was breaking due to the error and the fact that the 'branch' button was missing. Maybe the .git folder wasn't being created?
• Yep, I ran a clone on my local machine using the token to prove it was correct. And the user who owned the token had the correct access to the repo.
• Databricks seemed to be able to access its own workspace storage fine everywhere else. For example, my scripts which created folders within the instance's storage worked fine. And again, the repo was partially cloned.

I think maybe the fact that the accounts/principals used were 'brand-new' may have contributed to the error, perhaps? Given a few hours, it did eventually begin working. Perhaps Azure just needed to shake it out a bit.