I’ve been having the same problem since I get updated to macOS High Sierra 10.13.4
I have exactly the same issue. Insync will not sign in since High Sierra upgrade. I am trying using Chrome, Safari and Firefox as default browsers - I go through the process of granting access to google account, Insync web dashboard says I’m logged in my the app is not logged in. I click on it and it disappears. PLEASE FIX.