Shimon Peres definitely comes from lithuanina background – not Alexander chassidim.
Yitzchok Rabin and Ya’ir lapid are both from chassidic dynasties, And Rivlin, although more distanly connected is an tenth generation descendant of the baal hatanya. He actually spoke at a labuvitch ceremony once on yat teves in binyanei ha’oomah.
I think since many of our gedoilim were blessed with unusaul talents and/or tremendous determination and persistence it makes sense tha there fried out einiklach ended up much further in life.