I did find a solution that worked for me but forgot to post, but was reminded by a similar thread today, so I posted this solution there. I thought I'd better add it here in case others stumble on this thread, as I did originally.
I spent hours going through the permissions of the role that was meant to have the admin block and in the end it had nothing to do with permissions. The strangeness of the block's behaviour should have been a clue that the problem was with the block itself.
I found a post where Mary Cooch described the process for deleting and re-adding the admin block. I'll try to find a link to it but from memory the process is:
Go to your site front page, Turn editing on.
You need to unlock the admin block first so you can delete it. Go to:
Admin > site admin > plugins >blocks > manage blocks
Click the padlock icon for the Admin block so that it looks unlocked.
Go back to your site home page.
You should be able to delete the Admin block now. Take a deep breath and delete the Admin block.
Scroll down to 'Add block' and select Admin.
After a nervous wait you should have a fully functioning and well behaved Admin block.
Don't forget to go back in to manage block's and lock it again.
I hope this works for you as it did for me. If there are any Moodlerooms / Joule users reading this I can confirm it works there too (my problem was on a Joule site but I run a VM of the matching release of stock Moodle on my desktop for testing and comparison, plus sometimes it's just nice to have access to the code)
Thanks to Mary for her solution, it saved me at the time.