I uhh. What? Not really? Basically none of this checks out with what I've seen in the F/SN VN proper, all of NA F/GO, or what I can remember of F/HA. Where are you pulling all of this from?
Kara no Kyokai, Fate/EXTRA, F/SN Sasaki Kojirou is one of the Wraith Servants which is why he is summoned by Medea as she figures out the recursion trick to get a Servant of her own and bypass her status as a Caster to be a Master at the same time, Zouken Matou is himself a powerful Wraith at the point of F/SN but he doesn't qualify for the Assassin Vessel to his own satisfaction (High Presence Concealment and low everything else because he never trained himself to be good as a Servant and also never trusted another member of his family to be his Master or the caretaker of the Matou Magecraft), The Heaven Feel Fate Stay Night Movies (the giant mortar and pestle are animated in the third one), Jack the Ripper Assassin profile, I think there was something in the Unlimited Blade Works anime and I think I'm pulling stuff from Fate/complete material III.
Yeah, uh... none of that tracks. Its an established fact that Servants respond to the Grails call because they have a wish. Actual grail summoning mechanics are established as just a supercharged form of familiar magic - the classes come from needing to have some sort of limiter on the Heroic Spirits so that the Grail System can even manage them.
Their wishes have no bearing on them being summoned beyond meaning they're willing. We know this because of the one of thr most notable non-standard classes: Ruler.
Ruler's are specifically Heroic Spirits who DO NOT desire the grail for themselves IE no wish. The grail still summons them just fine.
The summoning was always about a wish with the Einzbern providing the alchemy that created the Greater Grail, the Tohsaka provided the land and the Vessels for the Servants and the Makiri provided the stabilizers to make sure the ritual didn't just blow up and the Command Spells to have a certainty of control over the Servants. This was for the First Holy Grail War which wasn't called that and happened in 1810.
The Second Holy Grail War was sometime in the 1860s and other than the name for the Ritual, the modification of the Servant's Vessels into something that was more limited by the Class system and the introduction of the Church as an arbiter, because the war between the Church and the Mage's Association that forced the Grail summoning to be done in Japan was done at that point, it didn't amount to much more than a murder spree.
In the Third Holy Grail War in 1939 the Eizberns did a cheating by creating their own Servant Vessel instead of using the 7 Standard ones to insure that the ritual would reach the Third Magic. In the F/SN timeline they created the Avenger Vessel and fucked up the ritual itself while in the Fate/Apocrypha they created the Ruler Vessel and almost unintentionally succeeded.
Anyways the Ruler Class is proof that there is a tether as it summons Servants that have no wish yet still have a desire to participate in the Living World as arbiters and protectors of it. It's nigh-impossible to complete a Ruler's tether as one would have to satisfy their internal need for helping people and/or the world.
Ah well, in any case.
Mordred! Mordred is fun.
Think Shirou can handle her?
Mordred nearly kills her master in the first five seconds of them meeting.
Oh boy. Depends. Which Class was she (he? I'm sorry both turn up in the fandom and it's been a while since I've seen/read Mordred in anything) been summoned into?
As far as I can tell Mordred isn't the usual Saber class so the options are Saber Alter, Berserker, Avenger or Beast (what with Mordred's far as I can tell/remember villainy being of the sort that would qualify her (him?) for a Beast Vessel).
Considering this segment:
Hatred.
It was… a good emotion, it kept you driven. It was all you understood these days. Hatred led to bloodshed, bloodshed led to death. Death… death led to everything you ever wanted, it was your goal. Your sole, driving purpose. For there were no others. There was no greater meaning in this life or any others, no heroes, no kings, no family.
There was only death, betrayal, and madness.
If you weren't fighting, what else was there? A pointless, shunned existence? A shadow, meant to perform their duty and little else?
No.
No you wouldn't accept that.
You would kill.
You would maim.
You would slaughter.
There is no meaning, not anymore. It is just mindless savagery, and they would all die. The innocent, the guilty, the loyal, the traitors, the strong, the weak. They would…
They would…
Die… by your hand.
Yes.
That's right.
Your hand…
But… who the fuck even were you, anyway?
Avenger or Beast would be my guess because you know Berserkers are still directed in their Madness and not this indiscriminate. So Shirou is in for either a perpetually hungry Taiga, only Mordred is hungry for murder instead of food, or an early Dark Sakura without the friendship/love Sakura has for him.
Either way Shirou is now suffering even more.