actually, I think that seeing the spaces and the intervals as you describe is incredibly important...for instance, if you see a 4-note chord written out with a note on every line, beginning with middle C, but ending with C instead of B, you instantly know that it's a C major chord, and you don't even have to look at the middle notes. you still need to be able to recognize any of those notes in isolation, but as you get better and better at sight reading, you will be able to look at any common chord without having to "spell" it. similarly, if you see a run of notes that begins with a B and goes stepwise down 5 notes, you don't have to spell it out, just do the run. Bach's music is full of examples of where this type of thinking will save you lots and lots of time.
when we learn how to read (print), we begin by learning letters and phonemes, and then grow to be able to recognize common whole words without having to "sound them out." this to me is an analogous process to what you are describing.