There is no inherent security problem with changing the content of the clipboard. That doesn't do anything until the user pastes it somewhere; of course if that "somewhere" is a command prompt, then that is a security problem, but users really ought to check what they're pasting there before they execute it (yeah, I know, "ought to").

It would be possible to do it the way you say, but that would mean that the user would need to allow that for many websites; I don't think copying from apps like Google Docs would work anymore, and "here's your access token, click here to copy it to the clipboard" features certainly wouldn't.

The screenshot in the OP would then probably be changed to include a step "click: allow clipboard access"; I think most people who fall for the screenshot in the OP would also fall for that.