Indeed, Shinji also has no desire that needs to be fulfilled by the Holy Grail.
When you think about it carefully, it's only natural. In the end, he is merely a passerby in this world. No matter how many traces he leaves behind or how deep the bonds he forms are, he will eventually have to leave.
After all, this is not the world he lives in, and the creation of the Third Magic cannot affect parallel worlds. Given that, why bother doing something pointless?
Of course, if there was a real chance to make a wish, Shinji wouldn't mind giving it a try.
Shinji and Justeaze's goal is to obtain the Holy Grail as a wishing machine, a sample of the miraculous magic power capable of triggering miracles, and to observe the entire process of converting that magic power into a wish.
It's not that they don't want to take the entire Greater Grail back with them, but they can't. As the old man of the jewel said, there are restrictions on traversing parallel worlds. Shinji can't bring his own body over because there is already a Shinji here. The same goes for the Greater Grail; there's an identical body of Justeaze in the mountain belly of Ryuudou Temple.
Therefore, the two can only settle for second best, using the incomplete Third Magic to carry back some untainted miraculous magic power. They'll also observe the critical operational mechanism of the Greater Grail to find a way to purify the tainted Holy Grail.
The miraculous magic power will only flow out once the Greater Grail is truly completed. Before that, it's merely the magical energy absorbed from the leylines, stored in the Greater Grail. For the magic power to be converted into a wish, someone needs to wish upon the Holy Grail and achieve it.
What, why not directly wish for the "method to purify it"?
This has already been explained earlier. The Greater Grail only shortens the process and gives the result, but it doesn't skip the process to directly produce the result. Without a method to achieve the wish, the wish cannot be realized.
Shinji and Justeaze don't know how to purify the Grail, nor do they know where the purification method exists. How could the Grail create one out of thin air?
If they were to make a wish, Shinji might wish for "Karna's full set of Noble Phantasms," "Gilgamesh's full set of Noble Phantasms," or "Ramses II's full set of Noble Phantasms"—wishes that are soul-bound and can be taken away.
But such wishes are not urgent. He is not a greedy person nor has Gilgamesh's collecting obsession. The Noble Phantasms at hand are sufficient for now. Getting those overpowered Noble Phantasms would be great, but not getting them isn't a big deal either. His mentality is somewhat akin to Kayneth's. This is precisely why he chose to participate as a Servant and then withdrew at a critical moment.
As for Fiore's wish... actually, this wish doesn't necessarily need the Holy Grail to be fulfilled. Compared to Kayneth's destructive magic circuit necrosis, Fiore's leg problem is a minor issue. For Justeaze, who has mastered 50% of Touko's medical skills, it's just a matter of surgery. If Gordes, the master of human alchemy, is willing to help, the treatment would be even smoother.
The reason for not mentioning it is that post-surgery recuperation requires a considerable amount of time, and it's not the right time to expose Justeaze. After the Holy Grail War ends, Shinji will naturally arrange for the surgery, allowing this lovely and intelligent girl to stand on the ground with her own feet, walk freely, and run.
Regarding the post-war handling of the Yggdmillennia clan's relationship with the Mage's Association, Shinji already has some preliminary ideas. If all goes well, the Yggdmillennia family will be able to break free from their losing streak alliance and truly secure a place in the Mage's Association. This way, Fiore's talents and potential won't go to waste, and their master-servant bond will not be in vain.
Fiore, even if I'm no longer here, you must live happily.
Although the mystery is declining, this era remains brilliant. I hope you will shine with your light—I will bless you from a distant parallel world.
◇◇◇
Just as Shinji thought he had accomplished a great feat and his thoughts were drifting away.
Shirou, the priest, walked out of the palace district with a katana in hand, stopping at a hideously jagged edge. Here lay the "Beast's Nest," which had been abandoned at his suggestion. From this vantage point, one could see the starry sky below and the earth above—yes, nothing seemed amiss. Semiramis's Noble Phantasm, the "Hanging Gardens of Babylon," was inverted, with its base facing the sky and its structures pointing toward the ground.
"The golden 'sun';, the silver moon, why do they look so familiar? Ah, I remember now, it's from there..."
Shirou squinted his eyes, exuding a mellow warmth like a jar of aged wine, unique and rich from years of fermentation. Unfortunately, no one was there to witness it.
"Unknowingly, so many years have passed. You who once amazed the world have gradually been submerged in the river of time. Besides historians who specialize in studying the past, probably only a few of us old immortals still remember you. Indeed, in any world, time is the most ruthless."
Shirou raised his head, his narrowly opened eyes firmly locked onto the "artificial sun."
"That's why I must obtain the Holy Grail. With this data, I can bring you out from the depths of history's river, to see the world you once dedicated yourselves to protecting. I can confidently say, this flourishing age is as you wished—no, it has become even better than you had hoped."
"I also want to show you the scenery beyond this world and ask for a bit of help. Waging war against a civilization that has plagued countless parallel worlds is a bit too much for just a few of us."
"So, I'm sorry. Although this 'surprise' you've given me is very satisfying, and I'm eager to see your next move, I still have to thwart your plans. I hope you don't mind."
"Well, even if you do mind, it doesn't matter. The world doesn't revolve around you, and things won't always go your way. True strength is finding a way out of adversity. Besides, after helping you so much, it's not too much to mess with you a bit occasionally. And this doesn't even count as messing with you."
At this, Shirou's mouth curled into a smile, and with his squinting eyes, he looked every bit the cunning old fox. He extended his left hand and softly chanted:
"Left Hand - Xanadu Matrix——Set!"
Red magical patterns flashed briefly, like a crimson lightning bolt shooting toward the tornado connecting the garden and the underground.
"So that's how it is, using the ley lines in reverse to channel magical energy to the Holy Grail, combining the ley lines and the Holy Grail into one entity. No wonder Assassin couldn't move it; it's the entire Trifas. But, there are ways to destroy it—"
He raised his right arm, with blue magical energy swirling around his wrist, ready to burst forth.
"——I just need to use these hands, no, it's still a bit too unfair to use them now, so... got it."
The blue light on his right hand faded.
"Lancer, release your power—let Brahmastra: O' Brahma, Cover the Earth."