This is just an overblown yellow-peril panic, right? How does any app paste? By "accessing the user clipboards". How does the chrome omnibox do "text you copied"? By "accessing the user clipboards".
I'd only refine your post to say that it's common for apps to read the clipboard without you pasting. Right click Chrome's omnibar and it will show "Paste and go to <clipboard contents>", my bittorrent client and RSS clients prepopulate the new torrent/feed form if I have a URL in my clipboard.
Is the tiny convenience worth the ability to snoop? I don't think so. Or rather, I would like to decide that for myself.
Now system API developers have to view downstream app developers in an adversarial manner...