When I click on the download symbol, my browser tries to open the file, presumably using the add-on that was installed when I installed Mathematica, but fails for the package I want.
Right-clicking the download tab in Windows gives a number of options, such as "save link as" but no download options. However, I also have a Mac, so I can use the process you describe.
On the Mac one can right-click and then specify where to download the linked file. I assume there's something similar for Windows if that's your OS.