I played this sonata in my senior year in college. I have an idea for you to consider...
When I was working on this, I made this funny little discovery that may or may not be worth mentioning, but I will anyway. It seemed to me that Mozart's piano sonatas seemed to fall into two categories - symphonies and operas. Hear me out...
The first movement of K570 could very easily be rescored as an opera. There are tutti sections, then places where one could easily hear a lead soprano, a basso buffo, or other solo voices playing around over simple orchestral accompaniment between interjected tutti sections where the 'whole orchestra' plays. When I started to think of the FIRST movement in this way a whole world of interpretation opened up for me. In contrast, K576 is NOT such a piece. 576 would transcribe into a very nice symphony.
The 3rd movement of K570 is not really like the first movement in this way, BUT the first movement colored my take on the 3rd. The key term I would apply to the 3rd movement is 'lyrical'. There are the little staccato sections that are like laughter and the entire movement, for me, was lighthearted and joyful. It might be your recording (sounds a bit overdriven and for that reason you might not have captured your dynamics in the recording), but I am not hearing the delicacy and nuance that a lot of these phrases can produce. Or, quite possibly, I am just hearing it differently than you. I would encourage you to think in terms of how a vocalist might perform many of the phrases in this movement and see if that brings any change to your concept of this work.
Hope you don't mind my comments. If it was not for my intimate knowledge of this piece I would have just said 'sounds good, keep up the good work'... ; )