Rather like when a given phrase can be said, using pauses, inflections, etc., to create a very different meaning, so a given list of pitches will give a very different melody when the same is applied. So, no, a list of pitches played in one order, but with different rhythms, could be used to produce thousands of different melodies. And that's without considering the different harmonies which could be used with that same set of notes - in the same order.
Then, there's the fact that music isn't constant sound. Rests play a big part - no sound is often as important as some sound.
Percussion. A big red herring. Leaving aside tuned percussion, there's the untuned, some of which make very short sounds (snare), some capable of long sounds (bells, cymbals), so they can easily be included in the 'horizontal'and'vertical' concept.
I don't think 'compete' is the word - 'complement' may be better. Rhythm and melody work together, horizontally and vertically. Taking one dimension away leaves us with no music as we know it.
Reading 'horizontally', it's way removed from reading words from a book. The length of pitches is pre-determined, which is of paramount importance to each melody. If that factor wasn't there, there would be a myriad of different ways to play a line of music. Certain 'rules' are usually observed, like partitioning the music into separate boxes (bars).
