It's weird that logging into the website temporarily fixes it. It's as if the device loses the connection to the server and can't reconnect, but yet visiting the website re-makes the connection.
Does anyone know if using the gmail app does the same thing, where it refreshes the connection on the stock mail app?
It sounds like an issue between Apple and Google and neither seem to be too keen to fix it. I'm also a google apps user.
For my critical account I am using the GMail app and it is working great.
For the other accounts, on Apple email, it is a hit / miss kind of relationship. This also applies to any emails I send via these specific accounts in the stock Apple email.
This behavior is the same on both my Mini 3 and 6S+.