The HDMI is 1080p and remains 1080p while recording, it's only not "clean" in the respect that it still shows the overlays. I doubt the live view will work at the same time, why would you want to look at that tiny screen if you have it hooked up to a monitor or EVF?
Because very often on a set you're not the one looking at the external screen. Often the camera operator wants to be looking at an EVF...which a viewfinder attachment can do a decent job of being mounted on the back of the camera body. A separate person--whether they are a focus puller, the director, the talent--will want to view the screen version. HDMI passthrough monitors allow multiple screens just fine as it is, but it would sure be nice to have the on-camera screen running simultaneously.
If Canon can't do this because of processing overhead constraints, OK, but a lot of practical filmmakers have demanded it like they have with many of the other features that were delivered.