Well... I uninstalled recent updates for the Google Search app, then the app appeared when I tried looking under "shortcuts" again. After doing this, I reinstalled the update...and the Google Search app was still under shortcuts. After I rebooted the phone however, the Google Search app was no longer under shortcuts...but my home screen shortcut still properly opens up the Google Search app - so I guess as long as I don't delete it I won't have any problems.
I think I just thought it was strange before how you can't add this shortcut via the app drawer...but could via long-tap > shortcuts.... It's also strange the recent update removed this option entirely - because I prefer this over the widget since it takes up 1/3 the space.