A test I did, a button that allowed me to load the site "in a window" (responsive design), in the full page.
I don't understand. How do you "add a button"?
If you want, you need to edit the css in Developer mode by Inspecting the source.
In desktop mode where the screen width is big, we can see all the menu such as Home, Products... Members, a searchbox, avatar+username, envelop and bell button. The "What's New" button (lightning icon) with badge is hidden. It is only visible in mobile mode.
In mobile mode, menu like Home, Products... are hidden in the hamburger menu so there are more space left to show more buttons on the top right menu.
I also not clear why the forum app wants to hide just 1 button in desktop mode.