That is the usual, but there are compositions, or more often parts of compositions, where the left hand carries the melody or a concurrent countermelody.
Der Fröhlicher Landmann by Robert Schumann (“The Merry/Happy Farmer” in English) is the one most piano students are first introduced to with this idea.
A number of parts of Liszt’s Hungarian Rhapsody #2 also show the melody in the left hand at times (like around 3:00).
The melody, or principal part in harmonized music, is typically based on human voice ranges. The Italians didn’t have many basso profundos and didn’t use it often and for the few works that were composed for them had challenges finding talent for performances. Russian Orthodox choral music uses them often like.
The typical guitar open tuning of E[sub]2[/sub]-E[sub]4[/sub] is basically in the typical Italian C[sub]2[/sub]-F[sub]4[/sub] range, and those open strings follow similar roles.
As a basso profundo myself I wish there was more in western music but yes it is pretty much always the melody higher and if you flip that you are stuck typically being viewed as a novelty act outside of Russian churches.
The same thing applies on the Guitar even if someone masters the skills to do so.
Interesting, rat avatar. Keep in mind, too, that the human brain “hears” the top line of a harmonized piece as the melody, unless the top line is sung or played at much lower volume than the lower line(s).