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According to the sheet music, the G minor chord and the octaves in the beginning of the piece should be split between both hands
And I see most people performing the piece like this as well (for example: Lugansky)
however, isn't this
much easier? The right hand wouldn't have to move at all, and the left hand only between G, B flat, D and A
One could play it as you suggest, and while it might be easier in the sense of jumping around the keyboard less, it would be a great deal harder musically.
The jumping around is what gives the piece its character. The lower octaves and higher repeated chords are like two armies at war with each other. It's much harder to achieve that effect if the physical effort of playing is smoothed over. It's harder to differentiate the two voices (the two armies, as it were) and it's harder to achieve the aggressive tone of the piece.
It's a lot harder to pull off the slur and legato in Bar 2 by playing its notes as left-hand octaves than by splitting its notes between the hands, especially if your hands can only span an octave each like mine.
At that point, you may as well similarly split up those low, exposed octaves in Bar 1.
Due to this section's "alla marcia" direction and the abundance of portatos in this edition of the measures, I don't think using pedal (to compensate for playing non-legato left-hand octaves) is appropriate.
