Why don't they do that, then? There must be some reason. I'm guessing it's that they want to override the content of what gets sent to the clipboard--is that correct?
I don't think normal web pages have access to the browser's clipboard. It's a security feature, I believe.
Normal web pages can't trigger copy/cut/paste events. They can access the clipboard via the events (which are automatically fired when the corresponding action is invoked by the user) documented in http://dev.w3.org/2006/webapi/clipops/.
You can buy another question is should you? I find the way Trello gets in my way very ill advised.