What I don't understand is how I can, allegedly (I've never tried), ask siri to dig through my photos and iMessages to look for stuff, but when I ask it to summarize the website on the screen it says it can only see literally what's on the screen and can't scroll down or grab the entire article that's already been loaded.

Those limitations come from a "protecting the user" mentality. That they know better what you should be allowed to do than you.