Fate/Silent Night

[Jorlem]

Saber only has access to NPs that she had on her death"bed", as she was summoned from there, not from the Throne of Heroes.  That's why she doesn't come equipped with Avalon, or Caliburn, or her spear Rhongomyniad, for example.  The Knights of the Round Table weren't present when she was summoned, so she couldn't have that as a Noble Phantasm when Shirou summoned her, unless something is significantly different about her situation from canon.

So, first, fuck Yuku and its broken quote functions. Seriously.

Anyway. It's well established that the power of belief distorts the abilities, appearance, and even behavior of the Servants. I don't think that a technicality like details of the timeline would stand between Saber and the single most fundamental point of her entire legend, especially given that other parts of it include the idea that King Arthur is 1, not dead but dreaming, and 2, that when 'he' rises again the Round Table will follow him.

Besides, as much as I do like to explore consequences, this is ultimately a crackfic, and readers are advised not to look at any given aspect too hard.

The power of belief can do that, once they've reached the Throne.  Arturia is being summoned from while she was still alive, so there wouldn't have been a chance for her to be altered by what was said about her later.  (And she's not going to the Throne in any case, she's going to go to Avalon.)  And at the time of her summoning, the Round Table was broken.  Galahad and Gawain were dead, and Lancelot, who was the most loyal, the 'ideal knight' to her 'ideal king', had committed treason.
And in any case, it was explicitly stated in the light novel of Fate/Zero that Arturia doesn't have the Knights of the Round Table as a NP:

Fate/Zero: Volume 2 wrote:

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...Even the horse of the King of Conquerors became an Heroic Spirit.
Apart from shock and admiration, everyone was speechless. Even
Archer, who also possessed a rank EX Noble Phantasm, was utterly silent
after seeing such a radiant army.

These heroes rode alongside their king on the battlefield; their wager, like the king’s, was the king’s dream.

Not even death could stop their ending loyalty. The King of Conquerors turned it into a fitting Noble Phantasm.

Saber was shaken to her core. It wasn’t the strength of the Noble
Phantasm that she was afraid of, it was the fact that such a Noble
Phantasm had de-stabilized her beliefs. It shook the beliefs that she
held in pride.

This flawless cooperation…

This bond with subjects that became a Noble Phantasm...

It was something that the idealistic King of Knights pursued for
her whole life, yet even to the end, it was something she could not
obtain.

Her issues with the Round Table are the reason for her intended wish in F/SN, as what she saw of, and was told by, Rider, combined with her encounter with Lancelot at the end of F/Z, where she witnessed the hatred he held for her, led her to believe that anyone could have done a better job as king than she did.
But yeah, as you said, this is just a crack fic, and I probably shouldn't be devoting this much time thinking about it.
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Stand between the Silver Crystal and the Golden Sea.
"Youngsters these days just have no appreciation for the magnificence of the legendary cucumber."  --Krityan Elder, Tales of Vesperia.