Why does solvent leveling occur?
Your final statement:


  in truth, whatever the $K_a$ it's not actually COMPLETELY dissociated


is exactly right.  

Strong acid is very often stated to be "completely dissociated", but that's just wrong.  

For example, nitric acid has a $K_a$ of 24, so 1M nitric acid about 96% dissociated. If the concentration is higher, it will be less dissociated and if the concentration is lower it will be more dissociated.  

I would think of the solvent leveling effect this way:

Suppose you have 1 liter of 0.1M ethanol in water and you add 0.1 moles of sodium amide thinking that it will deprotonate the ethanol.  That's not going to happen because most of the protons that protonate the amide will come from the solvent (water) rather than the ethanol.