This is a great idea. But from an implementation standpoint, it would be difficult to implement.
To explain: Our system stores a simple representation of a tweet. It doesn't know how to render it in html. Our server actually asks Twitter how to render the tweet based upon it's simplified representation (the twitter URL), using a service provided by Twitter. Twitter can change the html representation at any time. And at any given time there are many ways a Tweet can appear in rendered html.
When you paste raw html containing a tweet, or a YouTube or Facebook video, you are pasting the returned html and scripts from the provider, not the simplified representation we need to store it. The logic to go from this back to its simplified representation would be complex, and possibly subject to change at the whim of Twitter or these other providers.
But I'll think about this problem, I always like a good challenge. I certainly recognize that it would be very nice and convenient for our users if we could do this.