Shouldn't you match urls in the manifest, therefore it would only work on youtube pages? I don't see why you couldn't do this using Match Patterns. In any event, if you must choose between the background and popup, I would say popup.js because you would want it to check initially of clicking popup.html.
So you have popup.html, and then it has a button called "Activate". Just make another file called activate.js. In activate.js
function activate(){
var test = window.location.href
if (test.includes('watch?v')){
console.log('Yes')}
}
activate()
Include activate.js in your popup.html.