The European components (called, in Israeli discource, Ashkenazi) of the secular Israeli mainstream overwhelmingly trace their roots to Eastern Europe; Ben Gurion was born in Poland, Dayan's parents were from Ukraine, and the fathers of both Tzipi Livni and Bibi Netanyahu were born in Poland.
As Eastern European Jews secularized, many of them began to use non-Jewish languages for secular literature, while others (notably the more nationalist or socialist ones, e.g. the Bund or the Folkists) began using Yiddish as a secular language. However, those who went to Israel used Hebrew for these purposes, and those who stayed in Europe were killed.
Generally, it sounds like you're working under the misconception that the ultra-Orthodox and National Religious [1] are ethnically different from European-descended secular Israelis. They are not. They are distinguished by the ideological leanings of their grandparents in the early-20th-century, after which point individuals moving between the blocs became less and less frequent (though not unheard of).
[1] Very different groups, by the way - the National Religious uniformly speak Hebrew, and are most of the religious people in the settlements you're talking about.
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That's... not what Ashkenazi means at all.
The European components (called, in Israeli discource, Ashkenazi) of the secular Israeli mainstream overwhelmingly trace their roots to Eastern Europe; Ben Gurion was born in Poland, Dayan's parents were from Ukraine, and the fathers of both Tzipi Livni and Bibi Netanyahu were born in Poland.
As Eastern European Jews secularized, many of them began to use non-Jewish languages for secular literature, while others (notably the more nationalist or socialist ones, e.g. the Bund or the Folkists) began using Yiddish as a secular language. However, those who went to Israel used Hebrew for these purposes, and those who stayed in Europe were killed.
Generally, it sounds like you're working under the misconception that the ultra-Orthodox and National Religious [1] are ethnically different from European-descended secular Israelis. They are not. They are distinguished by the ideological leanings of their grandparents in the early-20th-century, after which point individuals moving between the blocs became less and less frequent (though not unheard of).
[1] Very different groups, by the way - the National Religious uniformly speak Hebrew, and are most of the religious people in the settlements you're talking about.