What is "the download icon"? I'm not even sure there is such a thing. My browser has an icon that looks little bit like "↓̲", so I suppose that’s what you might mean. Unfortunately, my poor web browser doesn’t render "↓̲" the same way my text editor did, so there is that.
If there is some frequently–used icon for downloads, you would have to convince everyone to use it, update all the fonts to include it, update all the keyboards to type it, etc. Meanwhile the rest of us will just be over here typing out "download" instead. The beauty of an alphabet is that anyone can compose a new sequence of letters and they automatically have a new word! Nobody has to ask permission, or convince anyone, or add characters to their fonts, or provide justification to the Unicode committee.
In principle you could add letters to the alphabet, but would that really be more efficient? Maybe words could start using fewer letters drawn from a larger set, but the storage requirements for a given any given piece of text would likely remain the same. After all, the computer already translates everything we type into binary, which is just an alphabet with only two characters. Each character that we type becomes 8 or more of these binary bits. Adding letters to the alphabet will just mean that we use more bits for every character. Plus there's still the problem of updating all the fonts, and keyboards, and textbooks, and teaching everyone about the new letters.
Meanwhile the rest of us can just compose the letters we have into new arrangements as often as we please. Perhaps you should search for "spelling reform" and read about all the past failures to change how we write before you jump to the conclusion that it would be an obvious improvement. See also "Shannon information theory".