The option to side with anyone doesn’t add anything inherently. Since the King resolves as either BD or Evil depending on who he’s siding with anyways.
Basically:
A: Neutral King sides with BD - Why not just spawn a BD King to the same effect?
B: Neutral King sides with Evil - Why not just spawn an Evil King to the same effect?
C: Neutral King sides with BD and then switches to Evil later - Now you have an Evil King that is immune to scumreading. Why? Because avoiding being scumread is actually pretty easy, the problem is that the most reliable way to do so is to peruse the town’s objective, which if you are scum is counterproductive even if you don’t get caught. However a Neutral King can afford to genuinely help BD and then switch to genuinely switching to Evil when he get’s the chance. An EK trying the same thing will also have to ensure that such a chance actually occurs and those attempts give something for the BD to scumread before it’s too late.
A Neutral King can afford act identically to a BD King, while an Evil King cannot.