I think that's a great idea. Make it more visible, and then you get normal people to care about it and put pressure on content owners, developers, etc. Like Google did with their "mobile-friendly" tag; I could yell at my managers all day about how we need to improve our site's mobile experience to no avail, but Google steps in and threatens lower rankings and suddenly it becomes top priority.
Perhaps the simplest solution is for Google to start penalizing heavy pages, but as far as I know, page weight isn't part of their mobile-friendly criteria.
I wonder what would happen if for example iOS decided to visually indicate page weight, kind of like how you can see which apps use the most energy.