How about I put it this way - by playing with even tones you lose your texture. At the moment, as I pointed out, I've got a 3 tune texture. That only exists because each 'tune' has it's individual characteristics.
That's a good ambition, however, this exercise is not written as several tunes, but as one melodic line. If he'd written it as several separate voices it'd be okay, but he didn't. Think of it as if you are playing a violin, playing a beautiful arpeggio. The more choices you have over how to play something the better, and being able to play it evenly will improve your control over more advanced textures too.
Actually now I have to practice it with only 2 'tunes' as I need to accent every 3rd note of the group:
That's better. Your fifth still looks a bit clumsy, you are kind of holding it away and then clawing at the keys instead of manipulating them. It looks like you press down too much with the arm on certain notes, this can be a big cause of unevenness. Try playing the exercise, but this time play it on your left arm instead of on the keyboard, it will help you sense which fingers you are pressing down on.