Why do timeouts require synchronized clocks?
Note that the authors also assume the following: 


  Crucial to our proof is that processing is completely asynchronous; that is, we make no assumptions about the relative speeds of processesor about the delay time in delivering a message.


The notion of "timing out processes" refers to the ability of knowing when to conclude that a process must have crashed. 
If you have a completely asynchronous system, then it does not help to equip processes with perfectly synchronized clocks, as these cannot be used to distinguish a slow process from one that has crashed.