Hello, Evernote backend engineer here. This issue was due to a couple of bugs on Google's side that are now solved.
https://issuetracker.google.com/issues/328859171
https://issuetracker.google.com/issues/328843663
The problem with the Drive Picker should now be solved on both Desktop and Mobile, without the need to upgrade Evernote. On web the Picker should have been functioning the whole time.
If you could confirm, that would be awesome! 😊