Thank you Steve (@sterndata)!
OK, I figured it out eventually. Just in case someone comes across this thread, you need to create a Google API key for your site. Go to the easy google fonts plugin, click on settings, then select the advanced tab. At the top right hand of the screen there is a help button. Click on that and you’ll see the option to click on Google API Key instructions. The plugin won’t display any google fonts without the API key, but it’s not obvious.
I’ve spent the past couple of hours trying to make this work and could not find anything. Help please!! 🙂 I can only get the webfonts to show in the typography customizer, none of the google fonts show up, unfortunately. Thanks so much!