Why is the King Air avionics switching system wired normally closed?
Design elements like this generally have to do with failure modes and how you want the system to fail when (not if) it fails. In this case, if there was a momentary or worse, permanent, loss of DC power to the relay or it failed for another reason you would want them to stay closed (and continue in the "ON") position so you would not lose your avionics (provided the power supply was some how still operational to them). Unlike most other electric components (semiconductors) relays are actually electro-mechanical components and they do have moving parts with in them that can fail. In my experience (non avionics mostly) relays have a much lower cycle lifetime than their semiconductor friends. 

If we dive even a bit deeper into them, a relay is effectively an electromagnet coil that actuates a small metal rod that generally moves a copper (or similar) strip to close a switch connect. Applying a steady state voltage to a coil to generate a magnetic field (and keep the relay in a desired position) also drives current through the coil. The coils, being very thin gauge heat, act like resistors and can generate a substantial amount of heat. Due to the very thin gauge of the wire this can lead to melting and shorting of the coil. The wire also tends to be resin coated which can heat and melt as well causing a short. In turn you are correct that they don't want to he help on for a long period of time. 

This is the electrical equivalent of many mechanically similar ideas in aircraft design. For example some landing gear in planes is held up using positive pressure so should the pumps fail the landing gear may simply deploy or be able to be shaken down by releasing a lock, the idea being its better it can come down in the case of a system failure than be stuck up.