Clipboard should be a private and secure enclave (encrypted) in the first place. Only the OS should store and retrieve data from it.
Without having the OS in charge of managing the entire UI, it's difficutlt to make that work
Aside from Clipboard managers (which should be handled like on screen keyboard or other alternative input methods) I don't see a reason to allow pasting (and so reading from the clipboard) from an window that doesn't have the input focus.
It's probably a possible problem to solve, but fixing stuff that wasn't build with security in mind from ground up is always tediuos.
https://developer.mozilla.org/en-US/docs/Web/API/Document/ex...
On the wider issue, I'd like to only allow a keyboard combination from the window manager / compositor to paste to the focused window. X11 has three different clipboards and I think Wayland has two (I've only seen xsel use the X11 secondary selection) while I've only ever heard of the one clipboard on Windows.
The kitty terminal has some method of using the clipboard that works remotely (though all the magic kitty does inspired me to "alias ssh='st ssh'"). I'm not sure if anything other than kitty uses it. The permissions have a specific "ask" option, which seems like something the browsers should support for the Clipboard API.