Here's a list of people who can't read sheet music:

Jimi HENDRIX
Django REINDHART (couldn't read at all actually)
Allan HOLDSWORTH
Slash
BB King
Louis ARMSTRONG
Robert JOHNSON

Well reading sheet music and understanding theory are two different things.
I personally read music pretty good, but my understanding of harmony and theory is way more developed (and useful for my job). I have picked that up when playing gigs, because you have to adapt to a singer's voice, etc.
To write lots of different songs, you need to know and understand theory, but your writing and reading chops don't need to be extreme, just sufficient to write what you hear in your head.