Yes, you DO want to be able to PASTE in something from your clipboard
to a web page, when convenient. But that's different than the web
page reading the contents of your clipboard directly whenever it wants
to.
As was mentioned, suppose the previous site you visited was someplace
you ordered something from and you copied and pasted in your credit
card number. Do you want every web page you access after that to be
able to read the clipboard and see your credit card number, even if
you're not pasting it into their web page?