The idea that redundancy in language is a sign of inefficiency is wrong. In fact it increases efficiency because it makes it less likely to misread or misinterpret something. I don't think you'd actually be increasing your bandwidth, if you remove superfluous letters your eyes won't have to look over them but you won't process the actual information any faster, you've just eliminated additional cues.

While it is possible to make a phonemic alphabet (phonetic would capture people's accents as well, which is probably not what you want), this removes the connection words have with their foreign origins. Plus pronunciation might change and you don't want to rewrite all books whenever that happens. You have to learn reading & writing anyway, so it might as well be a separate language to some extent, so that it can be more stable.