What actually is aggregate demand?
We measure these in values.

Also, this is a macroeconomic identity, it holds always by design.

You need an economic model that will make specific behavioral and other assumptions, in order to discuss movements when the price level change.

For example, you write


  it is far from intuitive that aggregate demand falls when the 'overall
  price level' (whatever that is) rises


Even in introductory economics courses beware of "intuitive" thinking.  

And indeed, where is the model and its assumptions based on which one would argue that as price level rises aggregate demand falls? It is certainly not embedded in the macroeconomic identity.  

Find the model that predicts so, understand its assumptions, and then discuss whether these assumptions are "intuitive".