All my piano life, I have been learning new songs mostly in the following "setup" (let's call it setup 1):
Left hand: Chords, arpeggios, small movements
Right hand: Melody, improvisations
Very rarely, however, have I encountered songs written the other way round (let's call it setup 2):
Left hand: Melody, improvisations
Right hand: Chords, arpeggios, small movements
Conclusion: The piano, as it is taught in many places in the world, is a concept by a majority of right-handed people.
However, to obtain, practice and maintain full dexterity in both hands, I would suggest that any and all piano lessons should be structured the following way:
Setup 1 - Setup 2 - repeat (50/50 balance)
(Later perhaps: two melodies, like in canons, etc. - this is just a suggestion for early practice)
Of course, you shouldn't break your arms, but it is perfectly possible to play bass lines as individual melodies with the left hand. In jazz songs, for example, I think it would be easy to switch hands because the accompanying chords of the voicings are very free, anyway, so you can rotate them any way you like.
The result of the current way of teaching piano is:
Most people, at a beginner or intermediate level, are much more agile in their right hands.
Only if they practice switching "chord hand" and "melody hand" will they practice "ambidexterity". I think that, especially for styles like jazz, this should give any piano students more flexibility in playing e.g. impros and increase overall piano "fluidity".
