I'm a bit confused about the don't backup to Google Drive but use Gmail.
are you trusting google or not?
are you trusting google or not?
But if you store the content of those messages with a cloud provider--regardless of which company--you have killed a lot of security value. The plaintext will sit there, just waiting for someone to go get it, and you'll never know if they do.
Email does not work that way. Even if it is encrypted in transit, it is always stored in plaintext in servers at either end. So, taking that into account, which email service provider has the best security team to defend that plaintext. This guide argues it is Google.