Even today, most Israeli Jews are Mizrahi Jews.
Yes and no. Mizrahim and Sephardim get clumped together as a single thing - mainly because a lot of Sephardim moved to mostly Mizrahi regions and because Mizrahim use Sephardic rites and customs - but they're not the same. You could say fairly that most Jews in Israel today are of Mizrahi-Sephardic descent, but that includes people of mixed Ashkenazi and Sephardic/Mizrahi background. Meanwhile Mizrahim is just an umbrella term meaning Eastern Jews (ergo, there are Yemenites, Syrian Jews, Baghdadi Jews, Persian Jews, Bukharan Jews, etc.). Many "Russian Jews" (ergo, Soviet or Russian Empire Jews) were Mizrahim (such as Bukharan Jews, or Turkic-speaking Krymchak Jews, or the Crimean Karaites) or 'other' (Georgian Jews, Mountain Jews of the North Caucasus and Azerbaijan, etc.). Many 'Russian Jews' are Ashkenazim from Ukraine. The labels are as messy as the history of the Jews as a people.
^That's a fairly accurate map, but also one with many flaws I could point out. Many Jews in France and the Netherlands were Sephardic emigre, for example. There were sizable Baghdadi Jewish communities in India over time. North African Jewry is associated with Sephardim, but there are many 'Berber Jews' who predated the arrival of the Sephardim.
EDIT: Fascinatingly, there's very little evidence of Roman-era conversions of Jews to Christianity (though there is some evidence of Samaritans who did so), but some Modern Palestinian and Negev Bedouin families are descended from Sephardim who fled the inquisition. The question of Crypto-Jews (not dissimilar, say, to the question of Crypto-Armenians in Eastern Turkey today) is an entirely different can of worms.
Israel's attempts to be inclusive to Jews of various overlapping but not mutually exclusive but also not identical definitions leads to weird politics that are really hard to place along usual ideas of left and right. You have a secular right wing Interior Minister, with the support of an Ethiopian Jewish centrist MK, who recently tried to put a cap on people who were half-Jewish or a quarter-Jewish ethnically migrating from Ukraine because the two figured the people of mixed Ukrainian and Jewish ancestry didn't 'count' for national or religious purposes. Alternatively, there's a quarter to half a million people in Israel who basically live as Jews, are of Jewish ancestry in part, but technically aren't Jewish because the Ultra-Orthodox have run the conversion system and won't include them (it's one of the biggest pressing issues for reform in the country). What's more important - ethnicity, religion, culture, patriotism, civic loyalty, etc.? Everybody's afraid of the country not being Jewish majority, but nobody can agree on what being Jewish even means. Meanwhile there's an ongoing process of 'Israelization' of the Arab community west of the wall, such that those folks are more interested in integrating into Israeli (not the same as Jewish) society overall, which is why the majority of Israeli Arab-speakers (because Arab-speaker is not the same as Arab, and Arab is not the same as Palestinian) voted for the moderate Ra'am party or for the Zionist parties in recent elections.