Sorry to reply so late, but seeing this thread and my take on this toggle menu which is the three lines in the upper left corner on mobile is that for desktops it's not there by design. This isn't a bug. If you have a large enough screen where the view is like the bottom picture you shared above, you can change the window size to smaller and you'll see it will switch to a mobile design at some point. This is nice for this type of situation as it shows that the designs are dynamic and not just tied to what type of device something is labeled, but that's another topic... bottom line is if I'm understanding you correctly the desktop design lacks the menu on the left that toggles on/off from the three lines.
Here is what I mean from taking two screenshots just now on my very large iMac screen, to highlight the window size can force the mobile version to turn on.
Now here is the same window but expanded to a normal size for my desktop.
Do you see the menu going away in the same way as your own devices? Any other details I'm missing potentially?
I'm not following completely... are you saying that you like the mobile toggle menu and would like a similar functionality on the full site?
If so these links and buttons are basically shifted to the notebook tabs on the full site. If you click on the down arrows you should be able to see a list of options. Here is what Forums looks like when clicked and the menu expands.