Magic in Tolkien’s Legendarium has two main parts: authority and secret knowledge. Secret knowledge allows those who have seen the light of the two trees to use the dual nature of the world to make artifacts that work in ways we can’t explain, and is effectively just science we can’t ever have access to. Anyway, you wanted to know about authority:

Looking at the example of Gandalf breaking the bridge, he says “I am a servant of the Secret Fire, wielder of the flame of Anor. The dark fire will not avail you, flame of Udûn. Go back to the Shadow! You cannot pass!” and then strikes the keystone with his staff. Let’s break this down:

1. Before we get to analyzing this line, we have to talk about his staff and his robes some more. These arent just clothes and a walking stick, these are his Uniform. If you went into a courtroom and the Judge was wearing jeans and a t-shirt, I dont think you would take them seriously. It’s really important to be dressed the part! The staff, in particular, functions like a badge of office. Think of all the stuff a cop can do by flashing their badge! They can’t do their job without it! The robes also signify his station so much he is named after his robes, but we arent sure why he is specifically grey.
2. First, he declares that he is a servant of the Secret Fire. This is the fire of creation, which belongs to and is part of Eru Iluvatar. This also has a dual meaning, since all mortals have a small piece of the Secret Fire inside them. So, with this line he is not only saying he serves God but also humanity.
3. By “wielder of the flame of Anor”, he could mean a few things. The Flame of Anor is the sun, literally speaking. I personally think he is invoking his ring of power, Narya, because i like taking things in the most literal way. The consensus among fans is that he is invoking his the power he gained as a servant of the the Lords of the West, since Anor was a fruit from the two trees which was placed in the sky by Nienna if memory serves. Nienna is significant here because Gandalf studied under her and learned from her the power of mercy, which is ultimately what defeats Sauron in the Lord of the Rings.
4. He addresses the balrog as the flame of Udun, and tells him that Morgoth will not save him here. “Go back to the shadow!” is at once a command to leave and telling him to go where Morgoth is, which is DEAD.
5. And then we get to the iconic “You Shall Not Pass!” line which is frankly better in the movies. He is not talking to the Balrog here. He is stating a new law of nature. It is now impossible for this balrog to cross this bridge.

This entire scene can be compared to a police officer showing up to a house, showing their badge and a warrant to search the place, announcing their intent to search the place, and then entering. Miss a step, and it doesn’t work.

Precedent has more to do with the powers that kings have. For example, Denethor’s line presides as the Steward of Gondor for so long the authority of his line seems comparable to Aragorn’s at some points, and this allows him to do some things that look like magic. One thing that comes to mind is that he seems to have a supernatural ability to spot falsehoods, which Sauron takes advantage of by feeding him half-truths. And let’s not forget this man held a Palantir in his hand for decades, stared Sauron in the face for decades, and he did not break. This man had a stronger will than Saruman!

There were arguably more examples of this variety of magic, where a king seems to have supernatural abilities because they are kings, and promises seem to have more power than laws. But if I detailed every little moment of magic i would be typing here all day.