Why do simultaneous-ops parallel runways need to be so far apart?
The simple answer seems to be for wake turbulence. This FAA document, ORDER JO 7110.308C, seems to say it is for wake turbulence reasons in para. 9:


  The geometry of the approach, as well as the lateral separation between the two approaches and prevailing local meteorological conditions, provide the wake turbulence avoidance necessary for reduced separation dependent approach operations.


Figure 3 shows the aircraft arriving in a staggered fashion, 1 nautical mile diagonal separation with runways 2500 ft apart (parallel runways separated by less than 2,500 feet, also referred to as, Closely Spaced Parallel Runways (CSPR)).  The text also describes how the lead aircraft follow a lower glide path than the following aircraft such that  the lead's wake turbulence isn't a factor for the following aircraft.