I'm not sure what you're arguing against. In the event that she decides to destroy the Dementors, which Harry anticipates to happen quite soon, she knows that the information she needs to be able to do so is already in her possession.

In that extremity, I went into the Department of Mysteries and I invoked a password which had never been spoken in the history of the Line of Merlin Unbroken, did a thing forbidden and yet not utterly forbidden.

It's possible that the Line reference is misleading, but if so it is an odd piece of phrasing.

Hence the example I suggest - whatever price the Unbreakable Vow exacts, there will be things that are worth it, like not going to Azkaban.

You mean like the fact that criminals can make Unbreakable Vows not to commit crimes, as an alternative to permanent trauma and probable death in Azkaban? (other criminals can power them for a reduction in sentence time or as part of the same type of bargain - permanent loss of some magical power is still better than Azkaban)

My opinion is that Harry's taught enough rationality at Hogwarts that the lie will fall apart.

Not judging by everyone's reactions when Hermione was accused of murder. A select few individuals might manage to question it at best.

Now that I think about it, shouldn't some Hogwarts students and/or teachers have figured out that they should be studying Muggle science? It's possible that this was mentioned, and I've forgotten it, or that (since EY probably won't write sequels, the subject will need to be left to fanfiction of HPMOR.

It hasn't been, except for Harry's suggestion in the latest chapter. Most people, though, still haven't been exposed to any noteworthy Muggle science except individual "clever tricks" as used by Harry - nothing to compare with the obvious power of magic.

In Chapter 46, he gives her a sealed note containing the explanation.

But if you ever need to fight Dementors, the secret is written here, cryptically, so that if someone doesn’t know it’s about Dementors and the Patronus Charm, they won’t know what it means...

"Phoenix's Price is the password that opens the stairwell to the room with broken wands, pictures and Pensieve vials. Phoenix's Fate is the password that opens the final door into that room. Both times Dumbledore takes Harry to the room, he speaks the first password, then the second.

I didn't say Harry will turn everyone into ponies

He's already partially responsible for turning Hermione into a unicorn, and Hermione is a prototype for how he wants all human beings to be (immortal and invulnerable). As long as he replaces or drops the Horcrux portion of the ritual, this seems like a realistic final ambition for him as far as means of defeating Death are concerned.

As a point of interest, wasn't it Merlin's original intent that, at minimum, everyone mentioned in a prophecy should have access to it? It was only centuries later that the Unspeakables sealed the prophecy records away, so why does the Line of Merlin Unbroken have a function for bypassing that seal, how does anyone know this, and why is using it forbidden?

"Crap," muttered Moody. His mad-eye was rolling wildly. "That's not good, not good at all."

(...)

"Crap," Moody said. Then Alastor Moody repeated, "Crap. Kid, should you even be saying this to us?"

"I don't know," Harry said. "If there's a user manual, I haven't looked at it yet."

"Crap."

I don't think Moody is trying to keep a poker face here.