Both of your answers assume there is a simple and easy way to identify what is "visible on screen" - which if you read through the OP, you will understand that is not an achievable concept.
I'm in favor of two-tier system where rich applications require permission for enhanced clipboard events but the average site will just copy what is selected without modification.
It's nowhere close to a solved problem, and is much more complex than this. Give a read through the comments in the OP for a taste of why.
Maybe there are deep reasons why it's impossible, but they are not well presented in the OP, unfortunately.
I suspect that for the great majority of practical cases, it is possible.
Where a definite answer is too hard to achieve, it's safe to err on the side of "not visible".
The bad guys aren't going to be using the great majority of practical cases.
The key principle of security is to deny all, allow a closed subset. Same here: allow what is obviously clear and has no hidden parts, and ask a confirmation for all the rest, legitimate or not.