Ah I was going to say you need to flow between the notes, however, I think the bigger issue here is rhythm and lack of continuity. There are just places where the music seems to freeze. Rubato isn't freezing the music; it's stretching it. When you freeze like that it seems like you are unsure whether you know the notes or not. It stops the flow of the music while if you stretch it, it just slows down the flow a bit.
And continuity: which is probably another huge reason why it sounds choppy. There's too much empty space between taking off the pedal and putting it back. I think you are taking the pedals too literally. Although it says to take it off on the 3rd note of each beat, I think you should hold it longer just so that the sound is continuous.