arfellows,
Have you been able to check the messages that are shown as 'unread' in the Mail app on each device to see if they have been read previously on one of the devices?
Be sure to also try logging into the webmail version of your Gmail account on their website, just to see what this shows regarding unread emails, as that may be a good indicator of what's happening server-side.
Just to note, there's a newer iOS version available as well. Updating will help ensure you have the latest fixes/improvements applied: Update your iPhone, iPad, or iPod touch - Apple Support
Similarly, try checking for any macOS updates as well: Update macOS on Mac - Apple Support
Cheers.