Makes sense to me

well mostly it's explained by the fact that it didn't happen.

"In rode the Lord of the Nazgûl, under the archway that no enemy ever yet had passed, and all fled before his face.

All save one. There waiting, silent and still in the space before the Gate, sat Gandalf upon Shadowfax: Shadowfax who alone among the free horses of the earth endured the terror, unmoving, steadfast as a graven image in Rath Dínen.

'You cannot enter here,' said Gandalf, and the huge shadow halted. 'Go back to the abyss prepared for you! Go back! Fall into the nothingness that awaits you and your Master. Go!'

The Black Rider flung back his hood, and behold! he had a kingly crown; and yet upon no head visible was it set. The red fires shone between it and the mantled shoulders vast and dark. From a mouth unseen there came a deadly laughter.

'Old fool!' he said. 'Old fool! This is my hour. Do you not know Death when you see it? Die now and curse in vain!' And with that he lifted high his sword and flames ran down the blade.

Gandalf did not move."

and then the Rohirrim arrive and the Rider leaves.

Dude really just "Go home, boy" the ghost.

Balrog who is also a Maia*

Maiar is plural

The answer is pretty simple:

Peter Jackson isn't nearly as good at telling a consistent story as Tolkien was. This scene isn't in the books.

I wouldn't say that. I'd say he did a great job adapting text to a visual medium. In the text just having the sword light on fire in preparation works. When you see it played out, everyone is going to ask why he didn't take a swing before leaving.
- This is exactly what the previous comment is saying. Jackson changed stuff for the movies to cater to mainstream movie audience expectations, at the cost of the worldbuilding and lore.

The Witch-King can fight a wizard for the same reason elves can fight and kill the greatest dragons and balrogs themselves. Binding creatures of spirit into physical form both weakens them and makes them vulnerable to other creatures built of both natures. If anything, as a wraith, he has an advantage in Tolkien rules.

That the Witch King is the equal of a balrog/Maia is not a stretch... Gandalf bring afraid of him even if he is a threat is the stretch, but I'd also argue that's not being shown in the scene.

Dragons weren't spiritual creatures bound into physical form. They were something Morgoth basically selectively bred.
- Fair enough in the sense that their exact origin isn't stated so I can't point to a passage and call you a damn casual. I will note that breeding them doesn't mean they didn't start as some kind of ainur or whatever.
  I'm curious how you think Ancalagon the Black is a pure product of animal husbandry though.

The sense is that Gandalf seemed too powerful and the director decided to humble him a bit to add tension.

And it was cool as fuck scene

LotR is running Pathfinder 2e under the hood, by the sounds of it, using Proficiency Without Level.

Could LotR scenes be used as examples to explain various mechanics from Pathfinder 2e?
- thegreatdarkness@ttrpg.network Sure. You should be able to use LotR to explain the rules of any fantasy RPG system, really.

Failed his will save too

Because Gandalf was a man, and the plot demanded it, perhaps?

He literally is not a man.
- Meaning both he and the Balrog lose a lot of their power in physical form in the first place but yeah