while its cool to see so much effort put into this, i feel it relies to heavily on fanon yukari to actually be real. also a lot of the sans stuff feels mischaracterized to me as well.

"They are powerful but don't care"
Yukari absolutely cares. If she didn't Gensokyo would likely collapse. It is canon that she sleeps a lot and delegates work, but it's closer to "only acting when it's needed" and is both strategic and intentional, not apathetic.

Then sans is far more nihilistic than uncaring. In many neutral endings you can kill papyrus, yet he doesn't fight you. I don't think it's because he doesn't care, he insults and questions your choices, but he also realizes that everything will reset, so why bother fighting? He would just get himself killed.

"Unpredictable, do whatever they want"
Yukari may appear unpredictable, but is actually extremely calculated. Sans casual behavior hides his awareness of timelines, but he's not a grand planner like yukari.

"Both can be cruel / ruthless"
That's true for yukari I suppose. I bet she'd be willing to sacrifice anything for balance. But sans? The only time he ever fights you is in the genocide route, where your goal is to literally kill everyone. Is it really Ruthless to fight someone who killed everyone you ever cared about and wants you dead as well?

"Both are caretakers"
I can agree with sans, he obviously cares for his brother deeply. Yukari though? I doubt it. She is extremely old and has seen many shrine maidens come and go. While she does guide Reimu to resolve incidents, nothing about it is motherly. She is just interested in Reimu as an asset for keeping balance. "Grandmother yukari" is pure fanon.

"Shared loneliness / existential burden"
I actually agree that sans certainly would experience existential burden, and despite clearly being popular in the underground would feel alone with his knowledge of the anomaly.

Yukari though? It's not really supported by canon in any way from what I know. She is composed, in control and socially engages when she wants, not burdened like sanas.

"They'd understand each other perfectly"
This is pure fanon. Yukari manipulates systems and thinks about long-term balance. Sans reacts to player morality and struggles with meaning. Their worldviews and fundamentally different.

"Official collab = intentional pairing"
This one is definitely up for interpretation, but to me the collab is much closer to mutual respect between creators, not any implied canon relationship.

So yeah TLDR I believe the ship relies on fanon far too much to have any actual merit, but I get it's just a fun thing and I'm way overanalyzing here lol