Why don't we use simple binary values to compress data?
Suppose we code, as you suggest, $a=0$, $b=1$, $c=10$, $d=01$, $e=11$, etc., and I send you the message $1001$. You can't tell if I said $10\,0\,1=cab$, $1\,0\,01=bad$. The point of the Huffman scheme is that it's a prefix code, so it's uniquely decodable (and, further, you can always decode the start of the message based on the bits you've already seen, without having to look into the future).