Verse 37:130

[Abu Muhammad]

سَلَامٌ عَلَىٰ إِلْ يَاسِينَ
إِنَّا كَذَٰلِكَ نَجْزِي الْمُحْسِنِينَ
إِنَّهُ مِنْ عِبَادِنَا الْمُؤْمِنِينَ

What does all these ayah means? Appoint Aali Yaseen as the leaders?

We will see who read the Satanic Quran and who isn't.

[Link]

سَلَامٌ عَلَىٰ إِلْ يَاسِينَ
إِنَّا كَذَٰلِكَ نَجْزِي الْمُحْسِنِينَ
إِنَّهُ مِنْ عِبَادِنَا الْمُؤْمِنِينَ

What does all these ayah means? Appoint Aali Yaseen as the leaders?

Yes that is implicitly implied.  It means Mohammad has a chosen family. And chosen ones are always appointed as leaders and guides.  And the theme was peace being upon exalted chosen souls.

[Link]

Peace be upon the family of Yaseen.

[Link]

The recitations are believed to reached us and to be believed in.

So the Surah before states 'Yaseen' and the Surah after resembles the opening lines in the way it recalls the Prophets of the past, but then ends up finishing it off with "Peace be upon the family of Yaseen."

It mentions other Prophets like Lut and Yonus (so not just Elyas) but finally blesses them with "And peace be upon the emissaries".

So for sake of argument even the recitation 'Elyaseen' is true, it would mean that God blesses the family of Yaseen with regards to Yaseen, in the same he blesses Elyas, Abraham, Nuh, etc, with regards to themselves.

So please don't say the family of Mohammad is not in Quran when they are according to the recitations that have been passed according to Sunni sources.