As long as there exists the possibility of mapping the 2 different domain name variants (with and without WWW) to different IPs or services, users need to see where they are.
Some domains are setup with WWW for serving a web site, and without WWW for access to email or the server software.
Every time there is a call from Google to obscure the reality of URLs for the sake of making them less obscure, I see an immediate vector open for misdirection and mischief.
We have enough trouble with clever use of encoded URLs and crafty use of Unicode.
Google want to hide everything not the central domain name, thus exposing users to higher risk of being on a fake site and not knowing.
Relying on a valid cert for authenticity is no use to newbies that have no idea where in the cert to look for the correct domain info.
Who issued the cert seems to be more important to display than which domains it is meant for (am I alone in seeing this as the wrong way round?)
Chromium doesn't even use proper OCSP so we have to put up with Googles crappy system (unless you enable it as a policy).
Ultimately this info is UI stuff so can be as optional as needed.
The full info should be on by default, and easy to change.
Personally I prefer to know exactly what I just trod in, so even favour seeing more info, such as a flag for geo-ip and the ability to see the IP of the domain I am on.