I just wanted to report back that I ended up solving my problem.
Fundamentally the issue was a user issue: I was trying to do something I hadn't done before thinking it was something I had done before.
What I have realised is that on my computer, the local OneDrive folder lives in the following location:
C:\dave\OneDrive
What I had been doing historically was copying the entire C:\dave folder which, because I have OneDrive set to download all files, meant I was able to back up the whole of my local OneDrive folder to my external hard drive.
The difference this time was I was trying to copy specific folders from within the OneDrive folder. Which is why I was running into a OneDrive message that I thought I shouldn't have been getting but was actually OneDrive operating like it has always operated.