What I occcasionally see is a “mobile site” which has the logo and a menu and not, as you say, the full site. We have found that if we don’t shuffle the posts each time (so 6 Featured, 3 Primary, 12 or less secondary (we usually go with the posts on the first page of posts in the back end and on page 2 set everything to Do Not Display) then eventually it will filter to the mobile view.
That being said it only seems to be an issue on iPhones. My sub editor using an Android has no viewing issues, and I can see it perfectly on an ipad.