Why would a Blue Angels pilot leave their gear down?
What possible reasons would justify leaving the aircraft in this state?

Because it's part of their show routine:


  
  Solos Take-off (Blue Angel #5: Dirty Roll on Take-Off; Blue Angel #6: Low Transition to High Performance Climb).
  


This dirty roll is not easy (it killed a Blue Angel back in 1982 in a Skyhawk during winter training), so it shows off skill and plane capability.

And later four planes fly gear down:


  
  Diamond Dirty Loop: The diamond flies a loop with all 4 jets in the carrier landing configuration.
  


The dirty loop formation started back in 1986 to celebrate the 40th anniversary and the new F/A-18:


  On 8 November 1986, the Blue Angels completed their 40th anniversary year during ceremonies unveiling their present aircraft, the McDonnell Douglas F/A-18 Hornet. The power and aerodynamics of the Hornet allows them to perform a slow, high angle of attack "tail sitting" maneuver, and to fly a "dirty" (landing gear down) formation loop.




Dirty is plane-speak for gear and/or flaps out, as opposed to a clean configuration.


Source: pinterest.com