Thanks for the suggestions. I think those 4 all look OK. The site content is purely static HTML (and some images) that hasn't changed significantly in months, so I'm guessing that's not the issue.
I'd rather not post the site URL here; is it OK if I email you?
How do the 4 SEO Tests look like? Outlined here in the "is it me or is it google" section https://medium.com/@franz.enzenhofer/what-to-do-when-a-googl...