I believe you're missing an important point: melody is not just a succession of notes, it is a timed succession.
As you pointed out, with the same "list" of notes you can get drastically different melodies depending on the duration of each note (and rests!) related to the others.
The horizontal/vertical orientation that is often used for harmony/melody is just a basic simplification that, in fact, is done to simplify explanation/communication.
In fact:
- harmony does follow and obey to rhythm too: besides the analysis of a specific "sum of sounds" at a given moment, the concept of harmony in a musical piece is based on the variation of those "sums" in a time context;
- harmony and melody are not distinct parts, they are closely related by their "vertical relations" and rhythmic relations;
- the "vertical" dimension is composed by both harmony and melody at a specific time;
Time is, in the end, the most important part of music, and for many reasons. Without time, the very concept of pitch (and, then, harmony) doesn't exist, since the pitch is the result of a frequency, which in turn is the result of a certain amount of events within a time frame.
Don't be confused by the concept of "percussion". While it generally refers to a clear sound event that happens from an absence of sound, it should in fact be considered more of a variation to the previous state: a change in frequency (pitch, timbre, harmonics, etc) and amplitude (dynamics). Consider that even a sound that follows a silence is a change in frequency: from zero (no sound) to X (some sound).
Our brains "simplify" most of those events, especially when their duration spans a certain amount of time (sustain, decay of sound, crescendo/diminuendo, etc), so the actual rhythm ("percussion") is perceived only when a drastic difference happens.
