Western music notation originated in religious vocal scores, at a time when church music was largely homophonic (just one melody). As melody is linear through time, using the horizontal axis to denote time makes a lot of sense - just like we do with the written word.

Of course Gradually tastes changed and Church music became polyphonic - adding a second (and third, fourth, fifth, etc) vocal line was a natural adaptation. Instrumentalists took it up outside of the Church, and the rest is history. We standardized on this notation because, even though it was originally for the human voice, it is convenient and good enough that everyone could make use of it.

Finally, I'll note that 20th century composers experimented endlessly with alternate notations, some more seriously than others: https://www.smithsonianmag.com/arts-culture/5-12-examples-of...