Intuitive explanation of the MA (mechanical advantage) of pulleys?

I think there's two key pieces needed to intuitively understand pulleys. The first is that, if you focus on the tension on the rope, you see that there are many ropes going between the pulleys, and just one going to the person doing the lifting. So, since the force of tension is the same all along the rope, the N ropes going back and forth between the pulleys generate N times the force.

The second intuition is that the puller and the heavy object are not the only actors in this game. The top pulley is attached to something. It's attached to some strong rig. You'll find that its the rig that provides the "missing" force that causes intuitive problems. While there may be only one rope being pulled on, all of the other tension forces holding up the heavy object pull down on the strong rig it is attached too. If that breaks, no number of pulleys will help you lift. Its that support that helps you sum all of the forces to equal 0.

The one thing you can't cheat is energy. Energy is conserved in a pulley system. So if you're pulling with 1/10th the force, you better be doing it over 10x the distance. And, indeed, if you work the math, that's what happens.

The total energy that you will end up spending is the same. The difference is that with a pulley you can use less force but more extended in time, so you can have a low power motor to raise a heavy weight. But it will take much longer with a smaller force, so the total energy spent will be the same.