Why does setting the Legacy 600's transponder to standby mode also deactivate its TCAS?
At least under FAA regulations the TCAS system goes into standby (by design) if the transponder fails


  A Mode S transponder is required to be installed and operational for
  TCAS II to be operational. If the Mode S transponder fails, the TCAS
  Performance Monitor will detect this failure and automatically place
  TCAS into Standby. The Mode S transponder performs the normal
  functions to support the ground-based ATC system and can work with
  either an ATCRBS or a Mode S ground sensor. The Mode S transponder is
  also used to provide air-to-air data exchange between TCAS-equipped
  aircraft so that coordinated, complementary RAs can be issued when
  required.


TCAS II systems get their altitude information from the transponder, which is a vital parameter to the system working. 

 

(source)

All modes also require a functioning Mode S transponder 


  
  Stand-by: Power is applied to the TCAS Processor and the Mode S transponder, but TCAS does not issue any interrogations and the
  transponder will reply to only discrete interrogations. The
  transponder still transmits squitters. Note: If the aircraft is on the
  ground and transmitting extended squitters, it is not required to
  transmit short (acquisition) squitters.
  Transponder: The Mode S transponder is fully operational and will reply to all appropriate ground and TCAS interrogations. TCAS
  remains in Stand-by.
  TA Only: The Mode S transponder is fully operational. TCAS will operate normally and issue the appropriate interrogations and perform
  all tracking functions However, TCAS will only issue TAs; RAs will be
  inhibited.
  Automatic or TA/RA: The Mode S transponder is fully operational. TCAS will operate normally and issue the appropriate
  interrogations and perform all tracking functions. TCAS will issue TAs
  and RAs when appropriate.
  


Strictly speaking all of this is only applicable to TCAS II systems but based on the noted case and my reading of the accident report it appears TCAS II was the system in use in this case.