I've used a third-party app on my phone for the last couple of years and every once in a while I get this same message. I get it, Google wants you using GMail so they can [a] feed you ads and [b] have more control over your device, but yes, the reminders are sort of cumbersome and can sound demanding. Sort of like when I access Gmail, G+ or any other Google service via Firefox or (yes, I use it occasionally) Edge and Google puts "warnings" all over the pages telling me their services work better in Chrome.
Now, don't get me wrong, Google makes some good services, but believe it or not, one size does not fit all. Google needs to learn this - or re-learn it.