Yabia, that’s really not true. Their leadership is and always was sefardi. Rac ovadia was the singular most accepted authority on all things sefardi, and he was the head for decades. Rav shalom cohen was a brilliant sefardi gadol too.
Shas built up sefardi yiddishkeit on a level not comparable to anyone else. Yeshivos, girls schools, infrastructure, everything; there’s almost no sefardi entity not directly or indirectly built and supported by shas members.
If by “Lithuanian charedi” you mean that they believe in the primacy of torah, and aren’t very zionistic, then you’re just describing what you personally feel sefardim should be.