I used to run my site with whatever.com as the main point for staff and existing customers, and basically administrative stuff, and www.whatever.com as the main public website. I had other hostnames as well, and the page lacking a hostname provided descriptive links to them as well. I don't do that anymore, but this change would make it confusing at best.

The stripped https:// isn't a problem since the lock symbol is there and only means that one-and-the-same thing.