Oh, I think I misunderstood something then. Sorry! (The documentation seems to suggest that API keys can be used here. Should I make a pull request to slightly change that part of the docs?)
Yes I can use SMTP but I am planning on using the same account across multiple sites (I’m cheap) and thought it would be cleaner to use an individual API key for each site, instead of the same SMTP user name / password across all sites.
then you can bypass SMTP entirely and use a WP plugin
But, but … I wanted to bypass plugins 
So what I have done is install the sendinblue plugin and given it the API key. It all works now.
Thank you for your help!