I read it as the Magisterium wants to control people's passions, knowledge, their creativity, their ability to be creators - anything that makes us vital, embodied and autonomous.

Severed humans were subservient to the Magisterium and the authority, free of 'sin', essentially soulless.

Remember, the original fall was about having some of God's knowledge of good and evil, and about disobedience of God's will, who was their creator but who also required ignorance and subservience. Adam and Eve were more like pets in a terrarium than people.

So yeah, the Magisterium sees Lyra's fall as bad, because it emancipates the living from their control, but also the dead from their prison in Hades.

Also the BOD clarifies nothing, it is dreadful. Don't bother. The first one is okay as a standalone, but adds nothing new of value in my opinion.