I also wonder about this. There used to be a Chrome extension you could use to analyze any page (logged in or not, taking into account all browser state), but that seems to be gone...
https://developers.google.com/speed/pagespeed/insights_extensions
My guess is that they got rid of it because various browser factors could affect performance in non-uniform ways.
That being said, it would be nice to be able to monitor pages which require authorization.