Let's handle this like we handle things like location tracking, notifications, etc: ask the user.

Except I don't think we should ask the user, because everyone will blindly click yes. I think we should disable this by default, and let Google Docs, etc., remind the user via some in-app modal with step-by-step instructions.

If someone actually needs this functionality, let them do the work to unlock it for a given website.

99.99999% of the web has no good reason to be modifying a user's clipboard.