Car interior light with multiple "triggers" - 12V DC power
The reason they wanted the ground of the LEDs  tied to the ground of the dome light is that most cars have 12 volts wired directly to the bulbs, and the cheap and simple door switch closes to ground (read chassis ground). I say most because my 2012 Mustang interrior lighting comes on gradually like a dimmer. Im guessing theres a simple PWM circuit in the "body control" computer that handles this function.... got to remember, a lot of old mechanics knows nothing about electronics.